Pack A — Fluency

Mathematics

Year 10 · Functions

Pack A · Fluency

Name: _________________________________

Date: _________________ Class: ___________

Answer all questions. Show your working. Questions are grouped by challenge level.

BronzeQuestions 1–10

1.
Given $f(x) = 3x + 2$ , find $f(4)$ .
2.
Given $f(x) = x^2 + 1$ , find $f(3)$ .
3.
Given $f(x) = 2x + 5$ , find $f(-3)$ .
4.
Given $f(x) = 3x + 2$ , find $x$ such that $f(x) = 17$ .
5.
A function maps $x$ to $2x + 3$ . Find the image of $5$ .
6.
A function maps $x$ to $3x - 1$ . What value of $x$ maps to $14$ ?
7.
A function $f$ has domain $\{1, 2, 3, 4, 5\}$ and rule $f(x) = 3x$ . List the range.
8.
Given $g(x) = x^2 - 4$ , find $g(0)$ and $g(3)$ .
9.
A linear function maps $0 \to 5$ and $1 \to 8$ . Find $f(x)$ .
10.
A function machine multiplies by $4$ then adds $7$ . Write the function $f(x)$ .

SilverQuestions 11–20

11.
State the domain and range of $f(x) = 2x + 3$ where $x \in \mathbb{R}$ .
12.
Find the range of $f(x) = x^2 + 3$ for $x \in \mathbb{R}$ .
13.
A function $f(x) = 3x + 1$ has domain $0 \leq x \leq 4$ . State the range.
14.
Given $f(x) = x + 3$ and $g(x) = 2x$ , find $(f \circ g)(4)$ .
15.
Given $f(x) = 2x + 1$ and $g(x) = x + 3$ , find $(f \circ g)(x)$ .
16.
Given $f(x) = x^2$ and $g(x) = x + 1$ , find (a) $(f \circ g)(3)$ and (b) $(g \circ f)(3)$ .
17.
Find the inverse of $f(x) = 3x + 2$ .
18.
Given $f(x) = 2x + 1$ , find $f^{-1}(9)$ .
19.
A function $f$ is defined by $f: x \mapsto 2x - 6$ . Find $f(5)$ and an $x$ with $f(x) = 0$ .
20.
A linear function $f$ satisfies $f(1) = 5$ and $f(4) = 14$ . Find $f(x)$ .

GoldQuestions 21–30

21.
Given $f(x) = 3x + 1$ and $g(x) = x^2$ , find $(f \circ g)(x)$ .
22.
Given $f(x) = 2x + 3$ and $g(x) = x^2$ , find $(g \circ f)(x)$ .
23.
State the largest possible domain of $f(x) = \sqrt{x - {a}}$ .
24.
State the largest possible domain of $f(x) = \dfrac{1}{x - {a}}$ .
25.
Given $f(x) = x^2$ and $g(x) = 3x + 4$ , solve $f(x) = g(x)$ .
26.
Find the inverse of $f(x) = \dfrac{x + {a}}{{b}}$ .
27.
Show that for $f(x) = 3x - 1$ and $f^{-1}(x) = \dfrac{x + {b}}{{a}}$ , we have $(f \circ f^{-1})(7) = 7$ .
28.
Given $f(x) = x + 3$ , $g(x) = 2x$ , $h(x) = x^2$ , find $(f \circ g \circ h)(2)$ .
29.
Given $f(x) = \sqrt{x}$ and $g(x) = x + 5$ , find $(f \circ g)(11)$ .
30.
Find the inverse of $f(x) = (x - 3)^2$ for $x \geq 3$ .

PlatinumQuestions 31–40

31.
Given $f(x) = 2x + 1$ and $g(x) = x + 3$ , find $(f \circ g)^{-1}(x)$ .
32.
Find the largest possible domain of $f(x) = \dfrac{1}{\sqrt{x - {a}}}$ .
33.
A car rental costs a £ $25$ fixed fee plus £ $12$ per day. (a) Write $C(n)$ for $n$ days. (b) Find $C^{-1}(c)$ and interpret.
34.
A function $f$ has graph $y = (x - 3)^2 + 2$ . Find the (a) minimum value and (b) range of $f$ .
35.
For $f(x) = 2x - 1$ and $g(x) = x^2 + 3$ , solve $(f \circ g)(x) = 13$ .
36.
A function $f(x) = ax + b$ satisfies $f(0) = 3$ and $f^{-1}(11) = 2$ . Find $a$ and $b$ .
37.
For $f(x) = x + 1$ , find $f(f(f(x)))$ .
38.
Given $f(x) = 3x + 2$ , find the value of $x$ for which $f(x) = f^{-1}(x)$ .
39.
A water tank has volume function $V(t) = 8t + 20$ litres after $t$ minutes ( $t \geq 0$ ). After how many minutes does the tank contain $100$ litres?
40.
For $f(x) = \sqrt{x}$ (domain $x \geq 0$ ) and $g(x) = x - 4$ , find the domain of $(f \circ g)(x)$ .