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A LevelMaths: PureAQAExam Questions2. Algebra & Functions2.8 Functions (A Level only)

Functions (A Level only) (AQA A Level Maths: Pure)

Exam Questions

3 hours33 questions
Easy
Medium
Hard
Very Hard
1
Sme Calculator4 marks

State whether the following mappings are one-to-one, many-to-one, one-to-many or many-to-many.

 

(i)
straight f colon space x rightwards arrow from bar 4 x minus 2
(ii)
straight f colon space x rightwards arrow from bar x squared
(
iii)
straight f colon space x rightwards arrow from bar x over 4
(iv)
straight f colon space x rightwards arrow from bar square root of x

 

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2
Sme Calculator3 marks

State the largest possible domains for the following functions.

(i)
straight f colon space x rightwards arrow from bar square root of x
(ii)
straight f colon space x rightwards arrow from bar ln left parenthesis x minus 2 right parenthesis
(iii)
straight f colon space x rightwards arrow from bar arcsin space x

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3
Sme Calculator3 marks

State the range for the following functions based on the given domains.

(i)
straight f colon space x rightwards arrow from bar e to the power of x               x element of straight real numbers
(ii)
straight f colon space x rightwards arrow from bar x squared space plus space 1        x element of straight real numbers
(iii)
straight f colon space x rightwards arrow from bar 1 over x               x element of straight real numbers

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4a
Sme Calculator3 marks

The function f(x) is defined as

straight f left parenthesis x right parenthesis space equals space x to the power of 2 space end exponent minus space 8 x space minus space 20 space space space space space space space space space space x element of straight real numbers

Sketch the graph of y = f(x), giving the coordinates of any points where the graph intersects the coordinate axes.

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4b
Sme Calculator2 marks

The minimum point on the graph of y = f(x) has x -coordinate 4.
Find the range of f(x) .

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5
Sme Calculator3 marks

The function f(x)  is defined as

straight f left parenthesis x right parenthesis space equals space x squared space space minus space space 9 space space space space space space space space space space x space greater or equal than space 3   

(i)
Work out the range of f(x).
(ii)
If the domain of f(x) is changed to x space greater or equal than space 0 comma what would be the new range of f(x)?

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6a
Sme Calculator4 marks

The functions f(x) and g(x) are defined as follows

 straight f left parenthesis x right parenthesis space equals space 3 x space plus space 5 space space space space space space space space space space x element of straight real numbers

g left parenthesis x right parenthesis space equals space minus 2 x space space space space space space space space space space space space space space space x element of straight real numbers space

Find

(i)
fg(x)   

(ii)
gf(x) 

 

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6b
Sme Calculator2 marks

Solve the equation f(x) = g(x) .

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7a
Sme Calculator3 marks

The function f(x) is defined by

 straight f left parenthesis x right parenthesis space equals space 3 x squared space plus space 1 space space space space space space space space space space straight x element of straight real numbers

Find the inverse of f(x) , f−1(x).

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7b
Sme Calculator2 marks

Find the domain and range for f-−1(x).

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8a
Sme Calculator3 marks

Solve the equation vertical line 6 minus 2 x vertical line space equals space 4.

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8b
Sme Calculator4 marks

On the same diagram, sketch the graphs of y space equals space vertical line 6 minus 2 x vertical line and y space equals space 4.
Label the coordinates of the points where the two graphs intersect each other and the coordinate axes.

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8c
Sme Calculator3 marks

Consider the graphs of y space equals vertical line 6 minus 2 x vertical line and y equals k, where k is a constant.
For which values of  k...

(i)
... will the two graphs have no points of intersection?

(ii)
... will the two graphs have one point of intersection?

(iii)
... will the two graphs have two points of intersection?

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1
Sme Calculator4 marks

State whether the following mappings are one-to-one, many-to-one, one-to-many or many-to-many.

(i)
straight f colon x rightwards arrow from bar x squared
(ii)
straight f colon x rightwards arrow from bar 3 x plus 1
(iii)
straight f colon x rightwards arrow from bar open parentheses x plus 1 close parentheses cubed
(iv)
straight f colon x rightwards arrow from bar plus-or-minus square root of x

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2a
Sme Calculator3 marks

The function straight f left parenthesis x right parenthesis is defined as

straight f left parenthesis x right parenthesis space equals space x squared space plus space 2 x space long dash space 3 space space space space space space space space space space space space x element of straight real numbers

Sketch the graph of y space equals space straight f left parenthesis x right parenthesis, giving the coordinates of any points where the graph intercepts the coordinate axes and the coordinates of the turning point.

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2b
Sme Calculator1 mark

Write down the range of straight f left parenthesis x right parenthesis.

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3a
Sme Calculator2 marks

The functionspace straight f left parenthesis x right parenthesis spaceis defined as

straight f left parenthesis x right parenthesis space equals space x squared space long dash space 4 space space space space space space space space space space x greater or equal than 0

Work out the range of straight f open parentheses x close parentheses.

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3b
Sme Calculator1 mark

If the domain of straight f left parenthesis x right parenthesis is changed to x space less or equal than space 0, what is the range of straight f left parenthesis x right parenthesis?

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4a
Sme Calculator1 mark

The functions straight f left parenthesis x right parenthesis and space straight g left parenthesis x right parenthesisare defined as follows

straight f left parenthesis x right parenthesis space equals space x squared space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space x element of straight real numbers g left parenthesis x right parenthesis space equals space 4 x space long dash space 3 space space space space space space space space space space space space space space space space space space space space x element of straight real numbers

Write down the range of straight f left parenthesis x right parenthesis.

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4b
Sme Calculator4 marks
Find 
(i)
straight f g left parenthesis x right parenthesis
(ii)
g straight f left parenthesis x right parenthesis

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4c
Sme Calculator2 marks

Solve the equation straight f left parenthesis x right parenthesis space equals space g left parenthesis x right parenthesis.

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5a
Sme Calculator3 marks

The graph of y space equals space straight f left parenthesis x right parenthesis is shown below.

q5a-1-2-8-medium-aqa-a-level-maths

(i)
Use the graph to write down the domain and range of straight f left parenthesis x right parenthesis.
(ii)
Given that the point (1, 1) lies on the dotted line, write down the equation of the line.

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5b
Sme Calculator2 marks

On the diagram above sketch the graph of y space equals space straight f to the power of negative 1 end exponent space left parenthesis x right parenthesis.

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6a
Sme Calculator3 marks

On the same axes, sketch the graphs of y space equals space straight f left parenthesis x right parenthesis and y space equals open vertical bar g open parentheses x close parentheses close vertical bar where

straight f open parentheses x close parentheses equals open parentheses x plus 2 close parentheses squared space space space space space space space space space space space space space space space space space space space space space space space space x element of straight real numbers straight g left parenthesis x right parenthesis space equals space 2 x space plus space 4 space space space space space space space space space space space space space space space space space space space space space space x element of straight real numbers

Label the points at which the graphs intersect the coordinate axes.

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6b
Sme Calculator2 marks

Solve the equation straight f left parenthesis x right parenthesis space equals open vertical bar g open parentheses x close parentheses close vertical bar.

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7a
Sme Calculator2 marks

The function straight f left parenthesis x right parenthesis is defined as

straight f colon x rightwards arrow from bar fraction numerator x squared plus 1 over denominator x squared end fraction space space space space space space space space space space space x element of straight real numbers comma space x not equal to 0

Show that straight f left parenthesis x right parenthesis can be written in the form

straight f colon x rightwards arrow from bar 1 plus 1 over x squared

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7b
Sme Calculator2 marks

Explain why the inverse of straight f left parenthesis x right parenthesis does not exist and suggest an adaption to its domain so the inverse does exist.

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7c
Sme Calculator4 marks

The domain of straight f left parenthesis x right parenthesis is changed to x space greater than space 0.

Find an expression for straight f to the power of negative 1 end exponent left parenthesis x right parenthesis and state its domain and range.

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8
Sme Calculator3 marks

Solve the equation open vertical bar 4 x space plus space 2 close vertical bar equals 5.

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9a
Sme Calculator3 marks

The functions straight f left parenthesis x right parenthesis and g left parenthesis x right parenthesis are defined as follows

straight f open parentheses x close parentheses equals 1 half open parentheses 4 x minus 3 close parentheses space space space space space space space space space space space space space space space x element of straight real numbers g open parentheses x close parentheses equals 0.5 x plus 0.75 space space space space space space space space space space space space x element of straight real numbers

Find
(i)
fg open parentheses x close parentheses
(ii)
gf open parentheses x close parentheses

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9b
Sme Calculator3 marks

Write down straight f to the power of negative 1 end exponent left parenthesis x right parenthesis and state its domain and range.

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1
Sme Calculator4 marks

State whether the following mappings are one-to-one, many-to-one, one-to-many or many-to-many.

(i)
straight f colon x rightwards arrow from bar 2 minus x cubed
(ii)
straight f colon x rightwards arrow from bar sin space x
(iii)
straight f colon x rightwards arrow from bar 1 over x squared
(iv)
straight f colon x rightwards arrow from bar In space x

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2a
Sme Calculator1 mark

It is given 

straight f open parentheses x close parentheses equals 2 over x

Write down the domain of the function text f end text left parenthesis x right parenthesis.

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2b
Sme Calculator3 marks

Sketch the graph of y space equals space straight f left parenthesis x right parenthesis, stating the coordinates of any intersections with the coordinate axes and the equations of any asymptotes.

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2c
Sme Calculator1 mark

Write down the range of straight f left parenthesis x right parenthesis.

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3a
Sme Calculator1 mark

The function straight f left parenthesis x right parenthesis is defined as

straight f left parenthesis x right parenthesis space equals space x open parentheses x plus 3 close parentheses squared space plus space 1 space space space space space space space space space space space space space x greater or equal than 0

Work out the range of straight f left parenthesis x right parenthesis.

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3b
Sme Calculator2 marks

If the domain of straight f left parenthesis x right parenthesis is changed to x space less or equal than space 0, what is the range of straight f left parenthesis x right parenthesis?

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4a
Sme Calculator1 mark

The functions straight f left parenthesis x right parenthesis and g left parenthesis x right parenthesis are defined as follows

straight f left parenthesis x right parenthesis space equals space 3 x squared space plus space 2 space space space space space space space space space space space space space space space x element of straight real numbers straight g left parenthesis x right parenthesis space equals space 1 space long dash space 3 x space space space space space space space space space space space space space space space x element of straight real numbers

Write down the range of straight f left parenthesis x right parenthesis.

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4b
Sme Calculator4 marks
Find
(i)
straight f g open parentheses x close parentheses
(ii)
g straight f open parentheses x close parentheses

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4c
Sme Calculator2 marks

Solve the equation straight f left parenthesis x right parenthesis space equals space straight g left parenthesis x right parenthesis space plus space 1

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5a
Sme Calculator3 marks

The graph of y space equals space straight f left parenthesis x right parenthesis is shown below.

q5a-2-8-hard-aqa-a-level-maths

(i)
Use the graph to write down the domain and range of straight f left parenthesis x right parenthesis.
(ii)
Write down the equation of the dotted line on the graph.

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5b
Sme Calculator2 marks

On the diagram above sketch the graph of y space equals space straight f to the power of negative 1 end exponent space left parenthesis x right parenthesis.

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6a
Sme Calculator3 marks

On the same axes, sketch the graphs of space y space equals space open vertical bar straight f open parentheses x close parentheses close vertical bar and y space equals open vertical bar straight g open parentheses x close parentheses close vertical bar where

straight f left parenthesis x right parenthesis space equals space 3 x space long dash space 1 space space space space space space space space space space space space space space space x element of straight real numbers g left parenthesis x right parenthesis space equals space 2 x space plus space 2 space space space space space space space space space space space space space space space x element of straight real numbers

Label the points at which the graphs intersect the coordinate axes.

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6b
Sme Calculator3 marks

Solve the equation = open vertical bar straight f open parentheses x close parentheses close vertical bar space equals space open vertical bar straight g open parentheses x close parentheses close vertical bar.

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6c
Sme Calculator1 mark

Which of the solutions to open vertical bar straight f open parentheses x close parentheses close vertical bar space equals space open vertical bar straight g open parentheses x close parentheses close vertical bar is also a solution to straight f left parenthesis x right parenthesis space equals space straight g left parenthesis x right parenthesis?

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7a
Sme Calculator3 marks

The functions straight f left parenthesis x right parenthesis and straight g left parenthesis x right parenthesis are defined as follows

straight f left parenthesis x right parenthesis space equals space e to the power of x minus 2 end exponent space space space space space space space space space space space space space space space space x element of straight real numbers g left parenthesis x right parenthesis space equals space 2 space plus space In space x space space space space space space space space x element of straight real numbers comma space x greater than 0

Find
(i)
straight f g left parenthesis x right parenthesis
(ii)
gf left parenthesis x right parenthesis 

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7b
Sme Calculator2 marks

Write down straight f to the power of negative 1 end exponent left parenthesis x right parenthesis and state its domain and range.

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7c
Sme Calculator2 marks

The graphs of straight f left parenthesis x right parenthesis and straight f to the power of negative 1 end exponent left parenthesis x right parenthesis are drawn on the same axes.

Describe the transformation that would map one graph onto the other.

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1
Sme Calculator4 marks

State whether the following mappings are one-to-one, many-to-one, one-to-many or many-to-many.

(i)
straight f colon x rightwards arrow from bar tan space x
(ii)
straight f colon x rightwards arrow from bar open vertical bar 1 over x close vertical bar
(iii)
straight f colon x rightwards arrow from bar square root of x squared end root
(iv)
straight f colon x rightwards arrow from bar plus-or-minus square root of 25 minus x squared end root

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2a
Sme Calculator2 marks

It is given

straight f left parenthesis x right parenthesis space equals space 4 x cubed space plus space 4 x squared space long dash space 7 x space plus space 2

Write down the domain and range of the function straight f left parenthesis x right parenthesis.

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2b
Sme Calculator3 marks

Sketch the graph of y space equals space straight f left parenthesis x right parenthesis, stating the coordinates of any intersections with the coordinate axes. (You do not need to give the coordinates of any turning points.)

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3a
Sme Calculator2 marks

The function straight f left parenthesis x right parenthesis is defined as

straight f left parenthesis x right parenthesis space equals space left parenthesis x space long dash space 3 right parenthesis squared left parenthesis x space long dash space 4 right parenthesis squared space space space space space space space 2 less or equal than x less or equal than 5

Work out the range of straight f left parenthesis x right parenthesis.

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3b
Sme Calculator1 mark

If the domain of straight f left parenthesis x right parenthesis is changed to x less or equal than space 2, what is the range of straight f left parenthesis x right parenthesis?

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3c
Sme Calculator1 mark

State another domain for straight f left parenthesis x right parenthesis that would have the same effect as that in part (b).

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4a
Sme Calculator1 mark

The functions straight f left parenthesis x right parenthesis and g left parenthesis x right parenthesis are defined as follows

straight f left parenthesis x right parenthesis space equals space x squared space minus space 2 space space space space space space space space space space space space space space space space space space x element of straight real numbers g left parenthesis x right parenthesis space equals space 1 minus begin inline style 2 over x end style space space space space space space space space space space space space space space space space space space space space x element of straight real numbers comma space x not equal to 0

Write down the range of straight f left parenthesis x right parenthesis.

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4b
Sme Calculator3 marks

Leaving your answers as single fractions, find

(i)   fg left parenthesis x right parenthesis
(ii)  gf left parenthesis x right parenthesis

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4c
Sme Calculator2 marks

Solve the equation straight f left parenthesis x right parenthesis space equals space straight g left parenthesis x right parenthesis

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5a
Sme Calculator4 marks

The graphs of y space equals space straight f left parenthesis x right parenthesis and y space equals space x (dotted line) are shown in the diagram below.
straight f left parenthesis x right parenthesis has rotational symmetry about the origin and for x space greater than space 0, there is a vertical line of symmetry at x space equals space 4.5.

q5a-2-8-very-hard-aqa-a-level-maths-pure

(i)
Use the graph to write down the domain and range of straight f left parenthesis x right parenthesis.
(ii)
On the diagram above sketch the reflection of straight f left parenthesis x right parenthesis in the line y space equals space x and explain why this cannot be the graph of straight f to the power of negative 1 end exponent left parenthesis x right parenthesis.

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5b
Sme Calculator3 marks
(i)
Given that the maximum solution to straight f left parenthesis x right parenthesis space equals space 6 is x space equals space 6, state the restriction on the domain of straight f left parenthesis x right parenthesis such that straight f to the power of negative 1 end exponent left parenthesis x right parenthesis exists.
(ii)
Hence, or otherwise, write down the domain and range of straight f to the power of negative 1 end exponent left parenthesis x right parenthesis.

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6a
Sme Calculator3 marks

On the same axes, sketch the graphs of y space equals space straight f left parenthesis x right parenthesis and y space equals open vertical bar g open parentheses x close parentheses close vertical bar where

straight f left parenthesis x right parenthesis space equals space square root of x space space space space space space space space space space space space space space space space space space space x greater or equal than 0 g left parenthesis x right parenthesis space equals space 2 x space long dash space 3 space space space space space space space space space space space x element of straight real numbers

Label the points at which the graphs intersect the coordinate axes.

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6b
Sme Calculator3 marks

Solve the equation straight f left parenthesis x right parenthesis space equals open vertical bar g open parentheses x close parentheses close vertical bar.

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6c
Sme Calculator1 mark

Which of the solutions to straight f left parenthesis x right parenthesis space equals open vertical bar g open parentheses x close parentheses close vertical bar is not a solution to straight f left parenthesis x right parenthesis space equals space g left parenthesis x right parenthesis?

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7a
Sme Calculator1 mark

The function straight f left parenthesis x right parenthesis is defined as

straight f colon x rightwards arrow from bar square root of 25 minus x squared end root space space space space space space space space space space space x element of straight real numbers comma space minus 5 less or equal than x less or equal than 5

Explain why the inverse of straight f left parenthesis x right parenthesis does not exist.

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7b
Sme Calculator2 marks

Suggest an adaption to the domain of straight f left parenthesis x right parenthesis so the following conditions are met:

  • the inverse of space straight f left parenthesis x right parenthesis spaceexists,
  • the graph of y space equals space straight f left parenthesis x right parenthesis lies in the first quadrant only, and,
  • the domain of straight f left parenthesis x right parenthesis is as large as possible.

State the range for your adapted straight f left parenthesis x right parenthesis.

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7c
Sme Calculator3 marks

The domain of straight f left parenthesis x right parenthesis is changed to negative 5 less or equal than space x less or equal than 0.

Find an expression for straight f to the power of negative 1 end exponent left parenthesis x right parenthesis and state its domain and range.

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8a
Sme Calculator3 marks

The functions straight f left parenthesis x right parenthesis and g left parenthesis x right parenthesis are defined as follows

straight f open parentheses x close parentheses equals open parentheses x minus 1 close parentheses squared minus 4 space space space space space space space space space space space space space space space space space x element of straight real numbers comma space space x greater or equal than 1 g open parentheses x close parentheses equals 1 plus square root of x plus 4 end root space space space space space space space space space space space space space space space space space x element of straight real numbers comma space space x greater or equal than negative 4

Find
(i)
straight f g left parenthesis x right parenthesis
(ii)
gf left parenthesis x right parenthesis

How did you do?

8b
Sme Calculator2 marks

Write down straight f to the power of negative 1 end exponent left parenthesis x right parenthesis and state its domain and range.

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8c
Sme Calculator2 marks

The graphs of straight f left parenthesis x right parenthesis and straight f to the power of negative 1 end exponent left parenthesis x right parenthesis are drawn on the same axes.
Describe the transformation that would map one graph onto the other.

How did you do?

8d
Sme Calculator2 marks

Find the coordinates of the point where the graphs of y space equals space straight f left parenthesis x right parenthesis and y space equals space straight f to the power of negative 1 end exponent left parenthesis x right parenthesis meet.

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9a
Sme Calculator4 marks

The graph of the function f left parenthesis x right parenthesis equals a vertical line x plus p vertical line plus q space spaceis shown below, where a comma space p space comma space q element of straight real numbers. The graph has a local maximum at the point A(3, 5)  and intersects the y-axis at (0,-7).

q7a-2-9-ib-aa-hl-further-functions-_-graphs-very-hard-dig

 

Find the values of a comma space p and q.Hence solve f open parentheses x close parentheses equals 0.

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9b
Sme Calculator4 marks

Find the solutions to the equation f left parenthesis x right parenthesis equals open vertical bar 7 minus 2 x close vertical bar.

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