Functions (Edexcel International A Level Maths: Pure 3)

Exam Questions

3 hours37 questions
1
Sme Calculator
4 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

Did this page help you?

2
Sme Calculator
3 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

Did this page help you?

3
Sme Calculator
3 marks

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

(i)

   space space straight f colon space x rightwards arrow from bar e to the power of x space space space space space space space space space space space space x element of straight real numbers

(ii)   

space straight f colon space x rightwards arrow from bar x squared plus 1 space space space space space space x element of straight real numbers

(iii)    straight f colon space x rightwards arrow from bar 1 over x space space space space space space space space space space space space space x element of straight real numbers

Did this page help you?

4a
Sme Calculator
3 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 minus 8 x minus 20         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 intersects the coordinate axes.

4b
Sme Calculator
2 marks

The minimum point on the graph of y space equals space straight f left parenthesis x right parenthesis has x-coordinate 4.
Find the range of straight f open parentheses x close parentheses.

Did this page help you?

5a
Sme Calculator
4 marks

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 equals 3 x plus 5      x element of straight real numbers                   

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

 Find     

(i)   straight f g left parenthesis x right parenthesis

(ii)   g straight f left parenthesis x right parenthesis

                 

5b
Sme Calculator
2 marks

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

Did this page help you?

6a
Sme Calculator
3 marks

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

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

Find the inverse of straight f left parenthesis x right parenthesisstraight f to the power of negative 1 end exponent left parenthesis x right parenthesis.

6b
Sme Calculator
2 marks

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

Did this page help you?

7a
Sme Calculator
3 marks

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

7b
Sme Calculator
4 marks

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

7c
Sme Calculator
3 marks

Consider the graphs of y 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?

Did this page help you?

1
Sme Calculator
4 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 x squared

(ii)     straight f colon space x rightwards arrow from bar 3 x space plus space 1 space

(iii)    

straight f colon space x rightwards arrow from bar left parenthesis x space plus space 1 right parenthesis cubed

(iv)
straight f colon space x rightwards arrow from bar space plus-or-minus space square root of x

Did this page help you?

2a
Sme Calculator
3 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 plus space 2 x space minus space 3 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 intercepts the coordinate axes and the coordinates of the turning point.

2b
Sme Calculator
1 mark

Write down the range of f(x) .

Did this page help you?

3a
Sme Calculator
2 marks

The function f(x) is defined as

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

Work out the range of  f(x) .

 

3b
Sme Calculator
1 mark

If the domain of f(x) is changed to x less or equal than 0 comma what is the range of f(x)?

 

Did this page help you?

4a
Sme Calculator
1 mark

The functions f(x) and g(x) are 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 x element of straight real numbers

straight g left parenthesis x right parenthesis space equals space 4 x space minus space 3 space space space space space space space space space space x element of straight real numbers           

Write down the range of f(x) .

 

4b
Sme Calculator
4 marks

Find

(i)      left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis

(ii)
left parenthesis g ring operator f right parenthesis left parenthesis x right parenthesis
4c
Sme Calculator
2 marks

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

Did this page help you?

5a
Sme Calculator
3 marks

The graph of y = f(x) is shown below.

 2-8-m-q5-edexcel-al-maths-pure

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

5b
Sme Calculator
2 marks

On the diagram above sketch the graph of y = f −1(x).

Did this page help you?

6a
Sme Calculator
3 marks

On the same axes, sketch the graphs of  y = f(x) and y space equals space vertical line g left parenthesis x right parenthesis vertical line  where

straight f left parenthesis x right parenthesis space equals space left parenthesis x space plus space 2 right parenthesis squared 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 4 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.

6b
Sme Calculator
2 marks

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

Did this page help you?

7a
Sme Calculator
2 marks

The function f(x) is defined as

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

 Show that f(x) can be written in the form

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

                             

7b
Sme Calculator
2 marks

Explain why the inverse of f(x) does not exist and suggest an adaption to its domain so the inverse does exist.

7c
Sme Calculator
4 marks

The domain of f(x) is changed to x > 0.
Find an expression for f−1(x)  and state its domain and range.

Did this page help you?

8
Sme Calculator
3 marks

Solve the equation  vertical line 4 x space plus space 2 vertical line space equals space 5.

Did this page help you?

9a
Sme Calculator
3 marks

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

straight f left parenthesis x right parenthesis space equals space 1 half space left parenthesis 4 x space minus space 3 right parenthesis 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 0.5 x space plus space 0.75 space space space space space space space space space x element of straight real numbers

Find

(i)
straight f g left parenthesis x right parenthesis
(ii)
g straight f left parenthesis x right parenthesis
9b
Sme Calculator
3 marks

Write down f−1(x) and state its domain and range.

Did this page help you?

10a
Sme Calculator
4 marks

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

straight f left parenthesis x right parenthesis space equals space vertical line x minus 9 vertical line 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 x squared space space space space space space space space space space space space space x element of straight real numbers                                        

Sketch the graph of y = fg(x) , stating the coordinates of all points where the graph intercepts the coordinate axes.

 

10b
Sme Calculator
2 marks

How many solutions are there to the equation fg(x) = 5 ?
How many solutions are there to the equation fg(x) = 9 ?

10c
Sme Calculator
2 marks

Write down the solutions to the equation fg(x) = 0.

Did this page help you?

1
Sme Calculator
4 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 space rightwards arrow from bar space 2 space minus space x cubed

(ii)     

straight f colon space x space rightwards arrow from bar space sin space x

(iii)

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

(iv)
straight f colon space x space rightwards arrow from bar space ln space x   

Did this page help you?

2a
Sme Calculator
1 mark

It is given

 straight f left parenthesis x right parenthesis space equals space 2 over x

                             

 

Write down the domain of the function f(x)

2b
Sme Calculator
3 marks

Sketch the graph of y = f(x), stating the coordinates of any intersections with the coordinate axes and the equations of any asymptotes.

2c
Sme Calculator
1 mark

Write down the range of f(x).

Did this page help you?

3a
Sme Calculator
1 mark

The function f(x) is defined as

 straight f left parenthesis x right parenthesis space equals space x left parenthesis x space plus space 3 right parenthesis to the power of 2 space end exponent plus space 1 space space space space space space space space space space x greater or equal than 0

Work out the range of f(x) .

3b
Sme Calculator
2 marks

If the domain of f(x) is changed to x less or equal than 0, what is the range of  f(x)?

Did this page help you?

4a
Sme Calculator
1 mark

The functions f(x) and g(x) 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 x element of straight real numbers

 g left parenthesis x right parenthesis space equals space 1 space minus space 3 x space space space space space space space space space space space x element of straight real numbers

Write down the range of f(x).

4b
Sme Calculator
4 marks

Find

(i)  fg(x)

(ii)  gf(x)
4c
Sme Calculator
2 marks

Solve the equation f(x) = g(x) + 1

Did this page help you?

5a
Sme Calculator
3 marks

The graph of y = f(x) is shown below.

28-q5-edexcel-al-maths-pure-hard 

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

5b
Sme Calculator
2 marks

On the diagram above sketch the graph of y = f−1(x) .

Did this page help you?

6a
Sme Calculator
3 marks

On the same axes, sketch the graphs of y equals vertical line straight f left parenthesis x right parenthesis vertical line  and y equals vertical line g left parenthesis x right parenthesis vertical line   where

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

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

6b
Sme Calculator
3 marks

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

6c
Sme Calculator
1 mark

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

Did this page help you?

7a
Sme Calculator
3 marks

The functions f(x) and  g(x) 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 space space x element of straight real numbers space space space

 g left parenthesis x right parenthesis space equals space 2 space plus space ln space x space space space space space space space space space space x element of straight real numbers comma x greater than 0

Find 

(i) straight f g left parenthesis x right parenthesis

(ii)g straight f left parenthesis x right parenthesis
7b
Sme Calculator
2 marks

Write down f−1(x)  and state its domain and range.

7c
Sme Calculator
2 marks

The graphs of  f(x)  and f−1(x) are drawn on the same axes.
Describe the transformation that would map one graph onto the other.

Did this page help you?

8a
Sme Calculator
4 marks

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

straight f left parenthesis x right parenthesis space equals space vertical line x minus 2 vertical line space minus space 5 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 vertical line x vertical line 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

Sketch the graph of y = gf(x), stating the coordinates of all points where the graph intercepts the coordinate axes.

8b
Sme Calculator
2 marks
(i)  

How many solutions are there to the equation gf(x) = 1?

(ii)  How many solutions are there to the equation gf(x) = 10?
8c
Sme Calculator
3 marks

Solve the equation gf(x) = 2.

Did this page help you?

9
Sme Calculator
3 marks

Solve the equation vertical line x to the power of 2 space end exponent minus space 4 vertical line space equals space 3 , giving your answers in exact form.

Did this page help you?

1
Sme Calculator
4 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 space rightwards arrow from bar space tan space x   

(ii)   straight f colon space x space rightwards arrow from bar space vertical line 1 over x vertical line

(iii)  straight f colon space x space rightwards arrow from bar space square root of x squared end root

(iv)  straight f colon space x space rightwards arrow from bar space plus-or-minus space square root of 25 minus x squared end root

Did this page help you?

2a
Sme Calculator
2 marks

It is given

straight f left parenthesis x right parenthesis space equals space 4 x cubed space plus space 4 x to the power of 2 space end exponent space minus space 7 x space plus space 2              

Write down the domain and range of the function f(x).

2b
Sme Calculator
3 marks

 Sketch the graph of  y = f(x), stating the coordinates of any intersections with the coordinate axes. (You do not need to give the coordinates of any turning points.)

Did this page help you?

3a
Sme Calculator
2 marks

The function f(x)  is defined as

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

Work out the range of f(x).

3b
Sme Calculator
1 mark

If the domain of f(x) is changed to x less or equal than 2, what is the range of f(x)?

3c
Sme Calculator
1 mark

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

Did this page help you?

4a
Sme Calculator
1 mark

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

straight f left parenthesis x right parenthesis space equals space x to the power of 2 space end exponent minus space 2 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 space minus space 2 over x space space space space space space space space space x element of straight real numbers comma x not equal to 0

           

Write down the range of f(x).

4b
Sme Calculator
3 marks

Leaving your answers as single fractions, find

(i)

  fg(x      

(ii) gf(x)

4c
Sme Calculator
2 marks

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

Did this page help you?

5a
Sme Calculator
3 marks

The graphs of y = f(x) and y = x (dotted line) are shown in the diagram below.

f open parentheses x close parentheses has rotational symmetry about the origin and for x greater than 0,there is a vertical line line of symmetry at x equals 4.5.

 2-8-vh-q5-edexcel-al-maths-pure

(i)
Use the graph to write down the domain and range of f open parentheses x close parentheses.

  

(ii)
On the diagram above sketch the reflection of f open parentheses x close parentheses in the line y equals x and explain why this cannot be the graph of f to the power of negative 1 end exponent open parentheses x close parentheses.

5b
Sme Calculator
2 marks
(i)
Given that the maximum solution to f open parentheses x close parentheses equals 6. is x equals 6 state the restriction on the domain of f open parentheses x close parentheses such that f to the power of negative 1 end exponent open parentheses x close parentheses exists.

 

(ii)
Hence, or otherwise, write down the domain and range of f to the power of negative 1 end exponent open parentheses x close parentheses.

Did this page help you?

6a
Sme Calculator
3 marks

On the same axes, sketch the graphs of y = f(x) and y space equals space vertical line g left parenthesis x right parenthesis vertical line 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 x greater or equal than 0

g left parenthesis x right parenthesis space equals space 2 x space minus space 3 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.

6b
Sme Calculator
3 marks

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

6c
Sme Calculator
1 mark

Which of the solutions to straight f left parenthesis x right parenthesis space equals space vertical line g left parenthesis x right parenthesis vertical line  is not a solution to f(x)  =  g(x) ?

Did this page help you?

7a
Sme Calculator
1 mark

The function f(x) is defined as

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

Explain why the inverse of  f(x) does not exist.

7b
Sme Calculator
2 marks

Suggest an adaption to the domain of f(x) so the following conditions are met:

  • the inverse of f(x) exists,
  • the graph of y = f(x) lies in the first quadrant only, and,
  • the domain of f(x) is as large as possible.

State the range for your adapted f(x).

7c
Sme Calculator
3 marks

The domain of f(x) is changed to negative 5 less or equal than 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.

Did this page help you?

8a
Sme Calculator
3 marks

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

 straight f left parenthesis x right parenthesis space equals space left parenthesis x space minus space 1 right parenthesis squared space minus space 4 space space space space space space space space space space space x element of straight real numbers comma x greater or equal than 1

g left parenthesis x right parenthesis equals space 1 space plus space square root of left parenthesis x plus 4 right parenthesis space end root space space space space space space space space space space x element of straight real numbers comma x greater or equal than negative 4

Find 

(i)  fg(x)

(ii)  gf(x)

8b
Sme Calculator
2 marks

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

8c
Sme Calculator
2 marks

The graphs of f(x) 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.

8d
Sme Calculator
2 marks

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

Did this page help you?

9a
Sme Calculator
4 marks

The graph of the function straight f left parenthesis x right parenthesis space equals space a vertical line x space plus space p vertical line space plus space 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 straight f open parentheses x close parentheses space equals space 0 space.

9b
Sme Calculator
4 marks

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

Did this page help you?

10a
Sme Calculator
3 marks

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

 straight f left parenthesis x right parenthesis space equals space vertical line x to the power of 3 space end exponent minus space 8 vertical line 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 open vertical bar x close vertical bar 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 space space space

Sketch the graph of y = fg(x), stating the coordinates of all points where the graph intercepts the coordinate axes.

10b
Sme Calculator
3 marks

There are between 0 and 4 solutions to the equation fg(x) = c , where c is a real
number.  Determine the values of c that produce each number of solutions.

Did this page help you?

11
Sme Calculator
3 marks

Solve the equation vertical line x squared space minus space 9 vertical line space equals space 6 space minus space 0.25 x squared, giving your answers in exact form.

Did this page help you?