CAIE IGCSE Additional Math 1 Functions Questions

[Maximum number: 10]

(a)

Figure for Question (a) — CAIE IGCSE Additional Math

Explain why this graph does not represent a function.

[ 1 ]

(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 ]

(c)

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

[ 5 ]

(i)

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

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

[ 2 ]

$\sin x \geqslant \frac{1}{2}$
May be implied. Allow >

a=150 seen

(ii)

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

[ 2 ]

$0 \leqslant \operatorname{gf}(x) \leqslant \frac{1}{\sqrt{2}}$

B1 for each correct value with correct sign associated with it

(iii)

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

[ 1 ]

No. Range of $\mathrm{g} \not \subset$ domain of g oe

[Maximum number: 5]

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

(a)

Write down the least possible value of a.

[ 1 ]

$-\frac52$ oe

(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 $\mathrm{g}(x)=x^{2}+1$ for $x \in \mathbb{R}$ .

[ 1 ]

$f\in\mathbb{R}$ oe

(c)

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

[ 3 ]

$\ln(2(x^2+1)+5)=4$ oe, soi
$2(x^2+1)+5=e^4$ or their $(2x^2+7)=e^4$ $x=\pm\sqrt{\frac{e^4-7}{2}}$ or exact equivalent
Alternative: solve $\ln(2x+5)=4$ to find $x=\frac{e^4-5}{2}$ , then state $g(x)=\frac{e^4-5}{2}$ .

[Maximum number: 10]

(a)

Table for Question (a) — CAIE IGCSE Additional Math

[ 5 ]

(b)

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

[ 5 ]

(i)

Write down the value of a.

[ 1 ]

(ii)

Write down the range of h.

[ 1 ]

$h(x)\geqslant3$

(iii)

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

[ 3 ]

$x=(y-1)^2+3$ $y=1+\sqrt{x-3}$ $h^{-1}(x)=1+\sqrt{x-3}$
Domain: $x\geqslant3$ . B1

[Maximum number: 5]

Table for Question 4 — CAIE IGCSE Additional Math

[Maximum number: 4]

The function f is defined by f(x)=2 x-1 for $x \in \mathbb{R}$ .

On the axes, sketch the graph of $y=\left|\mathrm{f}^{2}(x)\right|$ .

State any intercepts with the coordinate axes.

Figure for Question 5 — CAIE IGCSE Additional Math

$\mathrm{f}^{2}(x)=4 x-3$ soi
B1 for correct order of composition

Correct sketch with both intercepts indicated
FT their ( 4 x-3 ) providing B1 has been awarded and the expression is of the form m x+c

B1 FT for correct sketch with one intercept correct

[Maximum number: 4]

The functions f and g are defined as follows, for all real values of x.

$\begin{aligned}f(x)&=2x^2-1\\g(x)&=e^x+1\end{aligned}$

For each of the functions f and g , either explain why the inverse function does not exist or find the inverse function, stating its domain.

f is not one-one, hence $f^{-1}$ does not exist oe.
$g^{-1}(x)=\ln(x-1)$
M1 for $x=\ln(y-1)$ and a swap of variables at some point;
or $y=\ln(x+1)$ ;
or $e^x=y-1$ and $y=\ln x-1$ .

Domain: x>1. B1.

[Maximum number: 2]

It is given that $\mathrm{f}(x)=2 \ln (3 x-4)$ , for x>a, and that $\mathrm{f}^{-1}$ exists.

For your value of a, find an expression for $\mathrm{f}^{-1}(x)$ .

$\left[\mathrm{f}^{-1}(x)\right]=\frac{4+\mathrm{e}^{\frac{x}{2}}}{3}$ oe

M1 for a complete valid method,
allow a sign error
A1 must be using the correct notation

[Maximum number: 3]

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

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

$\ln(2(x^2+1)+5)=4$ oe, soi
$2(x^2+1)+5=e^4$ or their $(2x^2+7)=e^4$ $x=\pm\sqrt{\frac{e^4-7}{2}}$ or exact equivalent
Alternative: solve $\ln(2x+5)=4$ to find $x=\frac{e^4-5}{2}$ , then state $g(x)=\frac{e^4-5}{2}$ .

[Maximum number: 5]

function f is such that $\mathrm{f}(x)=2 x^{2}+5 x+3$ , for $x \geqslant p$ , where p is a constant.
It is given that $\mathrm{f}^{-1}$ exists.

Using your value of p, sketch the graphs of $y=\mathrm{f}(x)$ and $y=\mathrm{f}^{-1}(x)$ .
Label each graph.
State the intercepts of each of the graphs with the axes.

Figure for Question 9 — CAIE IGCSE Additional Math

Figure for Answer to Question 9 — CAIE IGCSE Additional Math

B1 for $y=\mathrm{f}(x)$ drawn as a one-one function in the first, second and third quadrants or first and second quadrants.

Must have correct curvature

Dep B1 for $y=\mathrm{f}^{-1}(x)$ as a reflection of their $y=\mathrm{f}(x)$ in the line y=x. Maybe implied by intercepts, but must have the correct curvature

B1 for $y=\mathrm{f}(x)$ clearly passing through x=-1 correctly soi. Allow if domain is unrestricted but must have correct quadratic shape or curvature

B1 for $y=\mathrm{f}(x)$ passing through y=3 only on the y-axis soi. Allow if domain is unrestricted but must have correct quadratic shape or curvature

B1 for - 1 and 3 marked correctly for $y=\mathrm{f}^{-1}(x)$ allow for unrestricted range.

Must have correct curvature.

CAIE IGCSE Additional Math 1 Functions Topic Practice

Covered syllabus points

CAIE IGCSE Additional Math 1 Functions Topic Practice

Covered syllabus points

Question 1

Question (a)

Question (b)

Question (c)

Question (i)

Question (ii)

Question (iii)

Question 2

Question (a)

Question (b)

Question (c)

Question 3

Question (a)

Question (b)

Question (i)

Question (ii)

Question (iii)

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9