Functions (Edexcel A Level Maths: Pure): Exam Questions

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Figure 1 shows a sketch of the graph with equation

Solve

Find .

State the range of .

Find , stating its domain.

The functions and are defined by

State the range of

Find

Find

The function is defined by

Find .

Show that where and are integers to be found.

Write in the form , where and are integers to be found.

Sketch the curve with equation showing any points of intersection with the coordinate axes and the coordinates of any turning point.

Find the range of the function

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

(i)     

(ii)     

(iii)    

(iv)

The function f(x) is defined as

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.

Write down the range of f(x) .

The function f(x) is defined as

Work out the range of  f(x) .

If the domain of f(x) is changed to  what is the range of f(x)?

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

Write down the range of f(x) .

Find

(i)      

(ii)

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

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

(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.

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

On the same axes, sketch the graphs of  y = f(x) and   where

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

Solve the equation .

The function f(x) is defined as

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

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

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

Solve the equation  .

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

Find

(i)

(ii)

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

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

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

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

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

Figure 4 shows a sketch of the graph of , where

Find the value of .

Find all the values of for which

The function is defined by

Explain why has an inverse but does not.

Solve

The function is defined by

Find

Show that

where and are constants to be found.

The function is defined by

State the range of

Find the range of

Figure 2 shows a sketch of the graph with equation

The vertex of the graph is at the point , shown in Figure 2.

Find the coordinates of .

Solve the equation

A line has equation , where is a constant.

Given that intersects at least once,

find the range of possible values of , writing your answer in set notation.

Figure 4 shows a sketch of the graph with equation

where is a positive constant.

Sketch the graph with equation where

stating

• the coordinates of the maximum point

• the coordinates of any points where the graph cuts the coordinate axes

Find, in terms of , the set of values of for which

giving your answer in set notation.

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

(i)      

(ii)

(iii)     

(iv)    

It is given

Write down the domain of the function f(x)

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

Write down the range of f(x).

The function f(x) is defined as

Work out the range of f(x) .

If the domain of f(x) is changed to , what is the range of  f(x)?

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

Write down the range of f(x).

Find

(i)  fg(x)

(ii)  gf(x)

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

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

(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.

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

On the same axes, sketch the graphs of   and    where

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

Solve the equation .

Which of the solutions to is also a solution to ?

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

Find 

(i)

(ii)

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

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.

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

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

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

(ii)  How many solutions are there to the equation gf(x) = 10?

Solve the equation gf(x) = 2.

Solve the equation , giving your answers in exact form.

The function is defined by

State the range of

Find

The function is defined by

Find

Find the exact value of the constant for which

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

(i)      

(ii)   

(iii)  

(iv)  

It is given

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

 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.)

The function f(x)  is defined as

Work out the range of f(x).

If the domain of f(x) is changed to , what is the range of f(x)?

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

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

Write down the range of f(x).

Leaving your answers as single fractions, find

(i)   fg(x)       

(ii) gf(x)

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

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

has rotational symmetry about the origin and for ,there is a vertical line line of symmetry at .

(i) Use the graph to write down the domain and range of .

(ii) On the diagram above sketch the reflection of in the line  and explain why this cannot be the graph of .

(i) Given that the maximum solution to . is  state the restriction on the domain of  such that  exists.

(ii) Hence, or otherwise, write down the domain and range of .

On the same axes, sketch the graphs of y = f(x) and  where

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

Solve the equation

Which of the solutions to   is not a solution to f(x)  =  g(x) ?

The function f(x) is defined as

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

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).

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

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

Find 

(i)  fg(x)

(ii)  gf(x)

Write down and state its domain and range.

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

Find the coordinates of the point where the graphs of  y = f(x) and  meet. 

The graph of the function is shown below, where . The graph has a local maximum at the point A(3, 5)  and intersects the -axis at (0,-7).

Find the values of  and Hence solve .

Find the solutions to the equation .

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

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

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.

Solve the equation , giving your answers in exact form.