1.2 Composite Functions

Quick Notes & Formulae

Essential concepts for solving composite function problems.

Definition

f g (x) = f (g (x))

. Apply

g

first, then

f

Order Matters

Generally

f g (x) \neq g f (x)

. Composition is not commutative.

Domain

Domain of

f g

is the set of

x

in domain of

g

such that

g (x)

is in domain of

f

Range

Range of

f g

is the set of outputs from

f

when inputs are restricted to the range of

g

Existence

For

f g

to exist, Range of

g

must overlap with Domain of

f

Past Paper Style Questions

Question 1: Basic Composition and Solving

The functions

f

and

g

are defined by:

f (x) = 3 x - 5

g (x) = x^{2} + 2

(a) Find

f g (x)

. [2]

(b) Find

g f (x)

. [2]

f g (x) \neq g f (x)

for all real

x

. [1]

(d) Solve

f g (x) = 10

. [2]

(e) Find all

x

such that

f g (x) = g f (x)

. [3]

Question 2: Square Root Domain

The functions

f

and

g

are defined by:

f (x) = \sqrt{x - 1}, x \geq 1

g (x) = 2 x + 3, x \in R

(a) Find

f g (x)

and state its domain. [3]

(b) Find

g f (x)

and state its domain. [3]

f g

. [1]

(d) Find the set of

x

for which

f g (x) \leq 4

. [2]

(e) Explain domain restriction necessity. [1]

Question 3: Rational Functions

f (x) = \frac{1}{x - 2}, x \neq 2

g (x) = x + 5, x \in R

(a) Find

f g (x)

and its domain. [2]

(b) Find

g f (x)

and its domain. [2]

f g

. [2]

(d) Solve

g f (x) = \frac{1}{2}

. [2]

Question 4: Modulus Function

f (x) = x^{2} - 4 x + 7

g (x) = | x - 1 |

(a) Find

f g (x)

. [2]

(b) Find

g f (x)

. [2]

f g (0)

and

g f (0)

. [2]

(d) Write

f g (x)

as piecewise. [3]

(e) State minimum value of

f g (x)

. [2]

Question 5: Logarithmic Function

f (x) = \ln x, x > 0

;

g (x) = x^{2} - 9

(a) Find

f g (x)

and its domain. [2]

(b) Find

g f (x)

and its domain. [2]

f g (x) = 0

. [2]

(d) Find the range of

f g

. [2]

Question 6: Rational Simplification

f (x) = \frac{2 x}{x + 1}

g (x) = x - 3

(a) Find

f g (x)

and its domain. [2]

(b) Find

g f (x)

and its domain. [2]

(d) Solve

f g (x) = 1

. [2]

Question 7: Reciprocal Function

f (x) = \sqrt{5 - x}, x \leq 5

;

g (x) = \frac{1}{x}, x \neq 0

(a) Find

f g (x)

and its domain. [3]

(b) Find

g f (x)

and its domain. [2]

Question 8: Mapping Notation

f : x \mapsto 2 x + 1, (- 2 \leq x \leq 4)

g : x \mapsto x^{2}, (0 \leq x \leq 3)

f g (x)

and its domain. [3]

(d) Find

g f (x)

and its domain. [3]

Question 9: Temperature Model

Sensor:

g (T) = 0.04 T + 1.2

. Display:

f (V) = 25 (V - 1)

(a) Find

f g (T)

. [2]

(b) Explain its meaning. [1]

T \in [- 10, 60]

. [1]

(e) Find the restricted domain. [2]

Question 10: Challenge Domain

f (x) = \frac{x - 1}{x + 2}

g (x) = \sqrt{x + 4}

(a) Find the domain of

f g (x)

. [3]

(b) Find the domain of

g f (x)

. [4]