Pack A — Fluency

Mathematics

Year 11 · 11.3 Functions

Pack A · Fluency

Name: _________________________________

Date: _________________ Class: ___________

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

BronzeQuestions 1–10

1.
Let $f(x) = 2x + 3$ . Find $f(5)$ .
2.
For $f(x) = 2x + 3$ , find $x$ such that $f(x) = 11$ .
3.
State the largest natural domain over $\mathbb{R}$ of $f(x) = \dfrac{1}{x - {a}}$ .
4.
State the largest natural domain over $\mathbb{R}$ of $g(x) = \sqrt{x - {a}}$ .
5.
Let $f(x) = x^2 + -3x + 2$ . Find $f(4)$ .
6.
A vertical line drawn through the graph of $y = x^2$ meets the curve in at most one point. Is $y = x^2$ a function of $x$ ?
7.
State the range of $f(x) = 2x + 1$ defined on $\mathbb{R}$ .
8.
For $f(x) = 3x - 2$ , find the image of $x = -2$ .
9.
State the largest natural domain of $h(x) = \dfrac{1}{\sqrt{x - 4}}$ .
10.
The function $f$ is defined by the table: $f(1) = 4$ , $f(2) = 7$ , $f(3) = 10$ . Find a formula for $f(n)$ if it is linear.

SilverQuestions 11–20

11.
Let $f(x) = 2x + 5$ and $g(x) = x^2 - 3$ . Find $f(g( 3))$ .
12.
Let $f(x) = 2x + 5$ and $g(x) = x^2 - 3$ . Find a simplified expression for $(f \circ g)(x)$ .
13.
Find $f^{-1}(x)$ for $f(x) = 2x + 5$ .
14.
State the domain and range of $f(x) = \dfrac{1}{x - 2}$ .
15.
For $f(x) = 2x + 5$ and $g(x) = -x + 8$ , solve $f(x) = g(x)$ .
16.
State the domain and range of $f(x) = |x - 3|$ .
17.
For what values of $x$ is $f(x) = \sqrt{2x - 6}$ defined?
18.
For $f(x) = x^2$ and $g(x) = x + 6$ , find all $x$ with $f(x) = g(x)$ .
19.
If $f(3) = 7$ and $f$ is one-to-one, state $f^{-1}(7)$ .
20.
Let $f(x) = \begin{cases} 2x + 1 & x < 0 \\ x^2 & x \geq 0 \end{cases}$ . Find $f(-3)$ and $f(2)$ .

GoldQuestions 21–30

21.
Let $f(x) = 2x + 5$ . Find $f^{-1}(x)$ and state its domain.
22.
Let $f(x) = \sqrt{x}$ and $g(x) = x - 4$ . State $(f \circ g)(x)$ and its largest natural domain.
23.
State the range of $f(x) = |x - 3| + 2$ .
24.
For $f(x) = \lfloor x \rfloor$ (the floor function), find $f(2.7)$ and $f(-1.4)$ .
25.
$f(x) = x^2$ is not invertible on $\mathbb{R}$ . State a domain restriction that makes it invertible, and give the inverse.
26.
A graph passes through $(0, 4)$ and decreases monotonically, approaching $y = 0$ but never reaching it. Is this consistent with $f(x) = 4 \cdot (0.5)^x$ ? Justify.
27.
Let $f(x) = 2x + 1$ and $g(x) = \dfrac{1}{x}$ for $x \neq 0$ . Find $(g \circ f)(x)$ and state its domain.
28.
For $f(x) = |2x - 4|$ , solve $f(x) = 6$ .
29.
The cost (CHF) of producing $n$ bottles is $C(n) = 0.5n + 200$ . State (a) the meaning of the gradient, (b) the meaning of $C(0)$ .
30.
The graph of $y = f^{-1}(x)$ is the reflection of the graph of $y = f(x)$ in which line?

PlatinumQuestions 31–40

31.
Let $f(x) = 2x + 1$ and $g(x) = \dfrac{x}{x - 1}$ for $x \neq 1$ . Find $(g \circ f)(x)$ and state its domain.
32.
For $f(x) = \dfrac{1}{x - 1}$ , find $(f \circ f)(x)$ and identify the values that are fixed by $f \circ f$ .
33.
Find the inverse of $f(x) = \dfrac{2x + 3}{x - 1}$ for $x \neq 1$ , and state its domain.
34.
State the largest natural domain of $f(x) = \dfrac{1}{\sqrt{x^2 - 4}}$ .
35.
For $f(x) = \begin{cases} 2x + a & x < 1 \\ x^2 & x \geq 1 \end{cases}$ , find $a$ such that $f$ is continuous at $x = 1$ .
36.
Find the range of $f(x) = x^2 - 4x + 7$ on $\mathbb{R}$ .
37.
The graph of $y = f(x)$ passes through $(1, 4)$ and $(3, 10)$ . If $f$ is linear, find $f^{-1}(x)$ .
38.
Let $f(x) = 2x - 1$ . Solve $f(f(x)) = 11$ .
39.
Let $f(x) = (x - 2)^2$ for $x \geq 2$ . Find $f^{-1}(x)$ and its domain.
40.
A circle of radius 5 centred at the origin has equation $x^2 + y^2 = 25$ . Explain why this is **not** a function of $x$ , and write the two functions $y = f_1(x)$ and $y = f_2(x)$ that together describe the circle.