• A Level Maths

Demo

• A Level
• Functions

1. 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

2. 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. $\{-2,\frac{2}{3},2\}$

3. A function is linear from $(-10,14)$ to $(-4,2)$, and from $(-4,2)$ to $(6,27)$. Solve $\mathrm{f}(x) = 12$

$x = -9, 0$

4. $$\mathrm{f}(x) = 2x^2 + 8x - 1$$ Find the largest possible domain for $\mathrm{f}$ such that it has an inverse.

$x \geqslant -2$ or $x \leqslant -2$

5. $$\mathrm{f}:x\mapsto x^2+2x-2$$ Find $\mathrm{f}:\{1, 2\}$

$\{1, 6\}$

6. Sketch the following graphs:

1. $y=|x-1|$
2. $y=|4x-3|$

7. Sketch the following graphs:

1. $y=|7-x|$
2. $y=-|x-1|$

8. Given $\mathrm{f}(x) = \dfrac{1}{x-2}$ and $\mathrm{g}(x) = 3x+4$, solve $\mathrm{gf}(x)=16$

$\dfrac{9}{4}$

9. Given $\mathrm{f}(x) = 5-2x$, solve $$\mathrm{f}^2(x) - (\mathrm{f}(x))^2 = 0$$

$\dfrac{6\pm\sqrt{6}}{2}$

10. $$\mathrm{f}(x) = \dfrac{x+6}{x+2}$$ $$\mathrm{g}(x) = 7-2x^2$$

1. Find the range of $\mathrm{g}$.
2. Solve $\mathrm{f}(x) = \mathrm{f}^{-1}(x)$.

1. $\mathrm{g}(x) \leqslant 7$
2. $x = -3, 2$

11. 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. $\mathrm{f}^{-1}(x) = \sqrt{x-5}-2, \quad x \geqslant 5$

12. Solve:

1. $|3x+1|=5$
2. $\left|\dfrac{x-5}{2}\right|=1$

1. $-2, \dfrac{4}{3}$
2. $3, 7$

13. Solve:

1. $\left|\dfrac{x}{6}-1\right|=3$
2. $|3x-5|=11-x$

1. $-12, 24$
2. $-3, 4$

14. Solve:

1. $3+\mathrm{e}^{2x}=7\mathrm{e}^{2x}$
2. $\mathrm{e}^{2x}=\mathrm{e}^x+2$

1. $-\frac{1}{2}\ln2$
2. $\ln 2$

15. 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 $x > \dfrac{13}{3}$

16. 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 $x \leqslant -13$

17. Solve:

1. $3|4x-3|-1=14$
2. $3-|2x+3|=1$

1. $-\frac{1}{2}, 2$
2. $-\frac{5}{2}, -\frac{1}{2}$

18. $$\mathrm{f}(x) = 2x^2-1, \quad x \geqslant a$$

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. $x=1$

19. 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 $\mathrm{fg}(7)$.

1. $6$
2. $\frac{21}{2}$

20. Solve:

1. $\ln(3x-2) - \ln 2 = \ln(x+4)$
2. $2\ln(x) = \ln(2x+3)$

1. $x=10$
2. $x = -1,3$

21. Given $$\mathrm{f}:x\mapsto x^2+3$$ $$\mathrm{g}:x\mapsto 2x+2$$ solve $$\mathrm{fg}(x) = 2\mathrm{gf}(x)+15$$

$3$

22. Solve

1. $3|x| + 5 = 5|x|-1$
2. $|4-x| = |4x-1|$

1. $\pm 3$
2. $\pm 1$

23. Solve:

1. $|2x-3| > |x+3|$
2. $|4x-1| < |x+1|$

1. $x < 0$ or $x > 6$
2. $0 < x < \dfrac{2}{3}$

24. 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. $-5 < x < \dfrac{1}{3}$