Problem-solving Pack

Mathematics

Year 10 · Functions

Problem-solving Pack

Name: _________________________________

Date: _________________ Class: ___________

These problems are designed to challenge you. Read each question carefully. Show all your reasoning — a correct answer without working receives no credit.

1Problem 1 of 12

Temperature converter. A function converts temperature from Celsius to Fahrenheit:

F(c) = \frac{9}{5}c + 32

.

(a) Find

F(0)

and

F(100)

.
(b) Find

F^{-1}(F)

, the inverse function (Fahrenheit to Celsius).
(c) Find

F^{-1}(212)

. Interpret.
(d) Find the temperature where Celsius and Fahrenheit are equal:

F(c) = c

Working space

2Problem 2 of 12

Function machines. Given

f(x) = 3x - 1

and

g(x) = x^2 + 2

:

(a) Find

f(2)

g(2)

f(g(2))

, and

g(f(2))

.
(b) Find a formula for

(f \circ g)(x)

and

(g \circ f)(x)

.
(c) Solve

(f \circ g)(x) = 14

Working space

3Problem 3 of 12

Inverse practice. Find the inverse of each function. State any domain restrictions needed.

(a)

f(x) = 5x - 7

(b)

g(x) = \dfrac{2x + 3}{4}

(c)

h(x) = (x - 2)^2

(for

x \geq 2

)

Working space

4Problem 4 of 12

Modelling a fence. A farmer has 60 m of fencing to enclose a rectangular field, one side of which uses an existing wall (so no fencing on that side).

If the side perpendicular to the wall has length

x

metres:

(a) Express the side along the wall in terms of

x

.
(b) Express the area

A(x)

as a function of

x

.
(c) State the domain of

A

.
(d) Find the value of

x

that maximises the area.

Working space

5Problem 5 of 12

Composition puzzle.

f

and

g

are linear functions with

f(x) = 2x + 1

.

If

(f \circ g)(x) = 4x + 7

, find

g(x)

Working space

6Problem 6 of 12

Identifying functions. For each relation, decide whether it represents a function. Justify.

(a) Each Year 10 student maps to their unique school ID.
(b) Each town maps to all citizens who live there.
(c) Each

x \in \mathbb{R}

maps to its square

x^2

.
(d) Each positive number

y

maps to all

x

with

x^2 = y

Working space

7Problem 7 of 12

Domain and range. A function is defined by

f(x) = \dfrac{1}{x - 3}

.

(a) State the largest possible domain.
(b) Find the range.
(c) Find

f^{-1}(x)

and its domain.

Working space

8Problem 8 of 12

Sketch and interpret. A function has graph passing through

(0, 4)

(2, 0)

(3, -1)

(5, 1)

(6, 4)

.

(a) Estimate the range from the data.
(b) Is the function one-to-one over

[0, 6]

? Justify.
(c) Why does this matter for finding an inverse?

Working space

9Problem 9 of 12

Currency converter. £1 = €1.18 (May 2026 rate).

(a) Write a function

E(p)

converting £

p

to euros.
(b) Find

E^{-1}

and interpret.
(c) Tomás is travelling from Geneva to London with €500. What does he have in pounds (2 d.p.)?

Working space

10Problem 10 of 12

Restricted-domain inverses. Consider

f(x) = x^2

on the full domain

\mathbb{R}

.

(a) Why does

f

not have an inverse on this domain?
(b) Restrict to

x \geq 0

. Now what is

f^{-1}

?
(c) Restrict to

x \leq 0

. Now what is

f^{-1}

?
(d) Verify (b): compute

(f \circ f^{-1})(9)

and

(f^{-1} \circ f)(3)

Working space

11Problem 11 of 12

Self-inverse functions. A function

f

is *self-inverse* if

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

for all

x

.

(a) Show that

f(x) = -x

is self-inverse.
(b) Show that

f(x) = \dfrac{1}{x}

(for

x \neq 0

) is self-inverse.
(c) Find all linear functions of the form

f(x) = ax + b

that are self-inverse.

Working space

12Problem 12 of 12

A composition mystery. Given

f(x) = 2x + 5

and

h(x) = 4x + 13

, find a function

g

such that

(f \circ g)(x) = h(x)

.

Also state whether

g

is unique. Justify.

Working space