Q1
Answer
Determine if the following functions are valid, and, if they are, whether they are one-to-one or many-to-one:
  1. $\mathrm{f}(x) = \pm\sqrt{x}$
  2. $\mathrm{f}(x) = \dfrac{1}{x}$
  1. one to many, not a function
  2. one to one, function
Q2
Answer
Find the range of the following functions:
  1. $\mathrm{f}(x) = x^2-5, \quad x \in \{-2,0,2\}$
  2. $\mathrm{f}(x) = \dfrac{2}{1-x}, \quad x \in \{-2,0,2\}$
  1. $\{-5,-1\}$
  2. $\{-2,\frac{2}{3},2\}$
Q3
Answer
A function is linear from $(-10,14)$ to $(-4,2)$, and from $(-4,2)$ to $(6,27)$. Solve $\mathrm{f}(x) = 12$
$x = -9, 0$
Q4
Answer
$$\mathrm{f}(x) = 2x^2 + 8x - 1$$ Find the largest possible domain for $\mathrm{f}$ such that it has an inverse.
$x \geqslant -2$ or $x \leqslant -2$
Q5
Answer
$$\mathrm{f}:x\mapsto x^2+2x-2$$ Find $\mathrm{f}:\{1, 2\}$
$\{1, 6\}$
Q6
Answer
Sketch the following graphs:
  1. $y=|x-1|$
  2. $y=|4x-3|$

Q7
Answer
Sketch the following graphs:
  1. $y=|7-x|$
  2. $y=-|x-1|$

Q8
Answer
Given $\mathrm{f}(x) = \dfrac{1}{x-2}$ and $\mathrm{g}(x) = 3x+4$, solve $\mathrm{gf}(x)=16$
$\dfrac{9}{4}$
Q9
Answer
Given $\mathrm{f}(x) = 5-2x$, solve $$\mathrm{f}^2(x) - (\mathrm{f}(x))^2 = 0$$
$\dfrac{6\pm\sqrt{6}}{2}$
Q10
Answer
$$\mathrm{f}(x) = \dfrac{x+6}{x+2}$$ $$\mathrm{g}(x) = 7-2x^2$$
  1. Find the range of $\mathrm{g}$.
  2. Solve $\mathrm{f}(x) = \mathrm{f}^{-1}(x)$.
  1. $\mathrm{g}(x) \leqslant 7$
  2. $x = -3, 2$
Q11
Answer
The function $\mathrm{f}$ is defined by $\mathrm{f}(x) = x^2+4x+9, \quad x\geqslant a$
  1. State the smallest value of $a$ for which $\mathrm{f}^{-1}(x)$ exists.
  2. Find $\mathrm{f}^{-1}(x)$ and state its domain.
  1. $-2$
  2. $\mathrm{f}^{-1}(x) = \sqrt{x-5}-2, \quad x \geqslant 5$
Q12
Answer
Solve:
  1. $|3x+1|=5$
  2. $\left|\dfrac{x-5}{2}\right|=1$
  1. $-2, \dfrac{4}{3}$
  2. $3, 7$
Q13
Answer
Solve:
  1. $\left|\dfrac{x}{6}-1\right|=3$
  2. $|3x-5|=11-x$
  1. $-12, 24$
  2. $-3, 4$
Q14
Answer
Solve:
  1. $3+\mathrm{e}^{2x}=7\mathrm{e}^{2x}$
  2. $\mathrm{e}^{2x}=\mathrm{e}^x+2$
  1. $-\frac{1}{2}\ln2$
  2. $\ln 2$
Q15
Answer
Solve:
  1. $|x-9| < 2$
  2. $\left|3-\dfrac{3x}{4}\right| > \dfrac{1}{4}$
  1. $7 < x < 11$
  2. $x < \dfrac{11}{3}$ or $x > \dfrac{13}{3}$
Q16
Answer
Solve:
  1. $|2x+9| > 14 - x$
  2. $-|3x+4| \leqslant 2x-9$
  1. $x > \dfrac{5}{3}$ or $x < -23$
  2. $x \geqslant 1$ or $x \leqslant -13$
Q17
Answer
Solve:
  1. $3|4x-3|-1=14$
  2. $3-|2x+3|=1$
<ol><li></li><li></li></ol>
  1. $-\frac{1}{2}, 2$
  2. $-\frac{5}{2}, -\frac{1}{2}$
Q18
Answer
$$\mathrm{f}(x) = 2x^2-1, \quad x \geqslant a$$
  1. Given that $\mathrm{f}^{-1}$ exists, suggest a suitable domain for $\mathrm{f}$.
  2. Find a solution to $\mathrm{f}(x) = \mathrm{f}^{-1}(x)$.
  1. $x\geqslant 0$
  2. $x=1$
Q19
Answer
The function $\mathrm{f}$ has domain $-5 \leqslant x \leqslant 14$ and is linear from $(-5,-8)$ to $(0,12)$ and from $(0,12)$ to $(14,5)$.
  1. Find $\mathrm{ff}(0)$.
  2. Given $\mathrm{g}:x\mapsto\dfrac{2x-5}{10-x}$, find $\mathrm{fg}(7)$.
<ol><li></li><li></li></ol>
  1. $6$
  2. $\frac{21}{2}$
Q20
Answer
Solve:
  1. $\ln(3x-2) - \ln 2 = \ln(x+4)$
  2. $2\ln(x) = \ln(2x+3)$
  1. $x=10$
  2. $x = -1,3$
Q21
Answer
Given $$\mathrm{f}:x\mapsto x^2+3$$ $$\mathrm{g}:x\mapsto 2x+2$$ solve $$\mathrm{fg}(x) = 2\mathrm{gf}(x)+15$$
$3$
Q22
Answer
Solve
  1. $3|x| + 5 = 5|x|-1$
  2. $|4-x| = |4x-1|$
  1. $\pm 3$
  2. $\pm 1$
Q23
Answer
Solve:
  1. $|2x-3| > |x+3|$
  2. $|4x-1| < |x+1|$
  1. $x < 0$ or $x > 6$
  2. $0 < x < \dfrac{2}{3}$
Q24
Answer
Solve:
  1. $6|x| \geqslant |2-3x|$
  2. $|x-3| > 2|x+1|$
  1. $x \geqslant \dfrac{2}{9}$ or $x \leqslant -\dfrac{2}{3}$
  2. $-5 < x < \dfrac{1}{3}$