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Exam-Style Questions.

Problems adapted from questions set for previous Mathematics exams.

1.

GCSE Higher

\(f(x) = \frac{2x}{5} + 7\) and \(g(x) = 10x^2 - 15\) for all values of \(x\).

Find \(fg(x)\).

Give your answer in the form \(ax^2 + b\) where \(a\) and \(b\) are integers.


2.

GCSE Higher

The function \(f\) is described by the following formula:

$$ f(x) = 3x^2 - 2x^3 $$

Calculate the value of \(f(-5) \)


3.

GCSE Higher

(a) A function is represented by the following function machine.

Function Machine

A number is input into the machine and the output is used as a new input.

If the second output is 53 work out the number that was the first input.

(b) A number is input into the machine and the output produced is the same number. Work out what this number could have been.

(c) Another function machine is shown below.

Function Machine

If the Input is 2, the Output is 7.
If the Input is 6, the Output is 27.
Use this information to fill in the two boxes.


4.

IGCSE Extended

If \(f(x)=5-4x\) and \(g(x)=4^{-x}\) then:

(a) Find \(f(3x)\) in terms of \(x\).

(b) Find \(ff(x)\) in its simplest form.

(c) Work out \(gg(–1)\) give your answer as a fraction.

(d) Find \(f^{–1}(x)\), the inverse of \(f(x)\).

(e) Solve the equation \(gf(x)= 1\).


5.

GCSE Higher

Here is a function machine that produces two outputs, A and B.

Function Machines

Work out the range of input values for which the output A is less than the output B.


6.

GCSE Higher

The functions \(f\) and \(g\) are such that:

$$ f(x) = 4x + 3 $$ $$ g(x) = x^2 - 3 $$

(a) Find \(f^{-1}(x)\)

(b) Given that \(gf(x) = 3fg(x)\), show that:

$$ 4x^2 + 24x + 33 = 0 $$

7.

IB Analysis and Approaches

The functions f and g are defined as follows:

$$f(x) = \frac{x+4}{5}$$ $$g(x) = 10x - 3$$

(a) Find \( (g \circ f)(x) \)

(b) Given that \( (g \circ f)^{-1} (a) = 5 \) , find the value of \(a\).


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The exam-style questions appearing on this site are based on those set in previous examinations (or sample assessment papers for future examinations) by the major examination boards. The wording, diagrams and figures used in these questions have been changed from the originals so that students can have fresh, relevant problem solving practice even if they have previously worked through the related exam paper.

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