CAIE IGCSE Mathematics Function Notation and Inverses

Practise evaluating functions, identifying domains and ranges, finding inverse functions and composing two or more function rules.

Exam points

substitute values or algebraic expressions into a function and simplify the resulting output
interchange x and y and rearrange to obtain an inverse function with valid notation
apply the inner function first in a composite such as fg(x), then simplify or solve the result

Covered syllabus points

E2.13.1 Understand functions, domain and range and use function notation.E2.13.2 Understand and find inverse functions f –1(x).E2.13.3 Form composite functions as defined by gf(x) = g(f(x))

[Maximum number: 10]

$10 \quad \mathrm{f}(x)=4-3 x \quad \mathrm{~g}(x)=x^{2}+x \quad \mathrm{~h}(x)=3^{x}$

(a)

Find fh(2).

[ 2 ]

23
M1 for $4-3(3^2)$ oe soi.

(b)

Find $f^{-1}(x)$ .

$f^{-1}(x)=$

[ 2 ]

$\frac{4-x}{3}$ oe final answer.
M1 for x=4-3y, y+3x=4, x+3y=4, $\frac{y}{-3}=\frac{4}{-3}+x$ oe, or $\frac{x}{-3}=\frac{4}{-3}+y$ oe.

(c)

Simplify.

[ 6 ]

(i)

f(1-2 x)

[ 2 ]

(ii)

$\operatorname{gf}(x)-9 \mathrm{~g}(x)$

[ 4 ]

20-36x or 4(5-9x) final answer.
B3 for 20-36x seen in working then spoiled.
OR M1 for $(4-3x)^2+4-3x-9(x^2+x)$ or better.
B1 for $(4-3x)^2=16-12x-12x+9x^2$ or better.
B1 for answer 20-kx or k-36x oe, or answer 20-36x with one sign/term error in working.

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Covered syllabus points

E2.13 Functions question 1

Question (a)

Question (b)

Question (c)

Question (i)

Question (ii)