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Pre-Calculus – Functions and Trigonometry

Polynomial, rational, exponential and trigonometric functions.

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12 questions  ·  36 marks  ·  answers included  ·  Pre-Calculus (US high school / AP preparation)
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This worksheet covers:

  • This worksheet covers polynomial
  • Rational
  • Exponential
  • Trigonometric functions

A calculator is allowed except where a question states otherwise, and you should aim to spend about 45 minutes on the full set.

Questions

Section A – Functions Skills

Calculator allowed
1

State the domain of $f(x)=\dfrac{x+3}{x^2-9}$.

Show answer
The denominator is zero when $x=3$ or $x=-3$, so the domain is all real $x$ except $x=3$ and $x=-3$.
[2]
2

Describe the transformation that maps $y=x^2$ to $y=(x-4)^2+1$. Hence state the coordinates of the vertex of $y=(x-4)^2+1$.

Show answer
Translation of $4$ units in the positive $x$-direction and $1$ unit in the positive $y$-direction. Vertex: $(4,1)$.
[2]
3

Given $f(x)=2x^3-3x^2-11x+6$:(a) Show that $x=3$ is a root of $f(x)=0$.(b) Hence factorise $f(x)$ fully.

Show answer
(a) $f(3)=2(27)-3(9)-11(3)+6=54-27-33+6=0$, so $x=3$ is a root.(b) Dividing by $(x-3)$ gives $2x^2+3x-2=(2x-1)(x+2)$, so $f(x)=(x-3)(2x-1)(x+2)$.
[4]
4

Find the equations of the vertical and horizontal asymptotes of $g(x)=\dfrac{3x-2}{x+5}$.

Show answer
Vertical asymptote: $x=-5$ (denominator zero). Horizontal asymptote: $y=3$ (ratio of leading coefficients, since numerator and denominator have equal degree).
[3]

Section B – Exponential, Logarithmic and Trigonometric Functions

Calculator allowed
5

Solve $5^{2x-1}=125$ algebraically.

Show answer
$125=5^3$, so $2x-1=3$, giving $x=2$.
[3]
6

Solve $\log_2(x+3)+\log_2(x-1)=5$, stating any restrictions on $x$.

Show answer
$\log_2[(x+3)(x-1)]=5 \Rightarrow (x+3)(x-1)=32 \Rightarrow x^2+2x-35=0 \Rightarrow (x+7)(x-5)=0$, so $x=-7$ or $x=5$. Since $x>1$ is required for both logarithms to be defined, $x=5$ is the only valid solution.
[4]
7

Solve $2\sin\theta-1=0$ for $0\le\theta<360^{\circ}$.

Show answer
$\sin\theta=0.5$, so $\theta=30^{\circ}$ or $\theta=150^{\circ}$.
[3]
8

Prove the identity $\dfrac{1-\cos^2\theta}{\sin\theta}=\sin\theta$ for $\sin\thetaeq0$.

Show answer
Using $\sin^2\theta+\cos^2\theta=1$, the numerator $1-\cos^2\theta=\sin^2\theta$. So $\dfrac{\sin^2\theta}{\sin\theta}=\sin\theta$, as required.
[2]

Section C – Challenge / Extended

Calculator allowed
9

A population of bacteria is modelled by $P(t)=200e^{0.35t}$, where $t$ is measured in hours. Calculate the time taken for the population to reach $1000$, giving your answer to 2 decimal places.

Show answer
$1000=200e^{0.35t}\Rightarrow e^{0.35t}=5\Rightarrow 0.35t=\ln5\Rightarrow t=\dfrac{\ln5}{0.35}=4.60$ hours (2 d.p.).
[4]
10

The height of a ferris wheel car above ground is modelled by $h(t)=15-13\cos\left(\dfrac{\pi}{6}t\right)$, where $h$ is in metres and $t$ is in minutes. State the maximum height, the minimum height, and the period of rotation.

Show answer
Maximum height $=15+13=28\,\text{m}$ (when $\cos=-1$). Minimum height $=15-13=2\,\text{m}$ (when $\cos=1$). Period $=\dfrac{2\pi}{\pi/6}=12$ minutes.
[3]
11

Solve $\tan^2x-3=0$ for $0\le x<\pi$, giving exact answers.

Show answer
$\tan^2x=3\Rightarrow\tan x=\pm\sqrt3$. In $0\le x<\pi$: $x=\dfrac{\pi}{3}$ (where $\tan x=\sqrt3$) and $x=\dfrac{2\pi}{3}$ (where $\tan x=-\sqrt3$).
[3]
12

A rational function has vertical asymptotes at $x=2$ and $x=-1$, a horizontal asymptote at $y=3$, and passes through the origin. Find a possible equation for $f(x)$.

Show answer
One possible answer: $f(x)=\dfrac{3x^2}{(x-2)(x+1)}$. This has vertical asymptotes at $x=2$ and $x=-1$, passes through the origin since $f(0)=0$, and as $x\to\pm\infty$, $f(x)\to3$ (equal degree numerator and denominator with leading coefficient ratio $3$).
[3]

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