CAIE IGCSE Additional Math 1 Functions Topic Practice

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Covered syllabus points

1.1. Understand function terminology1.2. Find domains and ranges of functions1.3. Use function notation1.4. Relate y = f(x) and y = |f(x)|1.5. Explain when inverses do not exist1.6. Find inverse functions1.7. Form and use composite functions1.8. Sketch functions and inverses

Question 1

[Maximum number: 10]

Question (a)

(a)
Figure for Question (a) — CAIE IGCSE Additional Math

Explain why this graph does not represent a function.

[ 1 ]

Question (b)

(b)

The table shows the graphs of four different functions.

Table for Question (b) — CAIE IGCSE Additional Math

Tick (✓)(\checkmark) each correct box in the table.
There may be more than one tick in a row or a column.

[ 4 ]

Question (c)

(c)

Functions f and g are defined as follows.
f: x↦sin⁡xx \mapsto \sin x \quad for 30∘⩽x⩽a∘30^{\circ} \leqslant x \leqslant a^{\circ}
g: x↦x−12x \mapsto \sqrt{x-\frac{1}{2}} \quad for x⩾12x \geqslant \frac{1}{2}
It is given that the function gf exists.

[ 5 ]

Question (i)

(i)

Find the value of a so that the domain of gf is as large as possible.

You may use the information that sin⁡30∘=12\sin 30^{\circ}=\frac{1}{2}.

[ 2 ]

Question (ii)

(ii)

For the domain found in part (i), find the range of the function gf .

[ 2 ]

Question (iii)

(iii)

Determine whether the function g2\mathrm{g}^{2} exists.

[ 1 ]

Question 2

[Maximum number: 5]

It is given that f(x)=ln⁡(2x+5)\mathrm{f}(x)=\ln (2 x+5) for x>a, where a is a constant.

Question (a)

(a)

Write down the least possible value of a.

[ 1 ]

Question (b)

(b)

Using your value of a, write down the range of f.

9 I given that 1(x)-1(2 x+5) or x>a, where a is a constant.
(a) Write down the least possible value of a.

It is also given that g(x)=x2+1\mathrm{g}(x)=x^{2}+1 for x∈Rx \in \mathbb{R}.

[ 1 ]

Question (c)

(c)

Using your value of a, solve the equation fg(x)=4.
Give your answers in exact form.

[ 3 ]

Question 3

[Maximum number: 10]

Question (a)

(a)
Table for Question (a) — CAIE IGCSE Additional Math
[ 5 ]

Question (b)

(b)

It is given that h(x)=(x−1)2+3h(x)=(x-1)^2+3 for x⩾ax\geqslant a. The value of a is as small as possible such that h−1h^{-1} exists.

[ 5 ]

Question (i)

(i)

Write down the value of a.

[ 1 ]

Question (ii)

(ii)

Write down the range of h.

[ 1 ]

Question (iii)

(iii)

Find h−1(x)h^{-1}(x) and state its domain.

[ 3 ]
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