Exam-Style Questions.

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.

Worked Solution

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2.	GCSE Higher

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

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

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

Worked Solution

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3.	GCSE Higher

(a) A function is represented by the following 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.

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.

Worked Solution

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4.	GCSE Higher

Four functions have been defined in the following way:

$$f(x) = 3x + 2$$ $$g(x) = 2x - 5$$ $$h(x) = \frac{1}{2x}, \, x \neq 0$$ $$j(x) = 3^x$$

(a) Find \(f(7)\),

(b) Find \(gf(3)\).

(c) Solve \(f(x) = g(x)\).

(d) Find \(f^{-1}(x)\).

(e) Find \(x\) when \(h(x) = j(-2)\).

(f) Write \(g(x) - h(x)\) as a single fraction.

Worked Solution

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5.	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 \(gf(x)= 1\).

Worked Solution

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6.	GCSE Higher

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

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

Worked Solution

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7.	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:

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

Worked Solution

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8.	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 \(a\).

Worked Solution

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