(i)

g(-1) = $\frac{\mathrm{1\; +\; 3(-1)}}{\mathrm{(-1)\; -\; 1}}$

g(-1) = $\frac{\mathrm{1\; -\; 3}}{\mathrm{-1\; -\; 1}}$

g(-1) = $\frac{-2}{-2}$

g(-1) = 1

(ii)

$\frac{\mathrm{1\; +\; 3x}}{\mathrm{x\; -\; 1}}$ = 7

Get rid of the fraction by multiplying both sides by the denominator (x - 1)

1 + 3x = 7 x (x - 1)

1 + 3x = 7x - 7

1 + 7 = 7x - 3x

8 = 4x

Divide both sides by 4

x = $\frac{8}{4}$ = 2

(ii)

g(-3) = $\frac{\mathrm{1\; +\; 3(-3)}}{\mathrm{(-3)\; -\; 1}}$

g(-3) = $\frac{\mathrm{1\; -\; 9}}{\mathrm{-3\; -\; 1}}$

g(-3) = $\frac{-8}{-4}$

g(-3) = 2

(a)

f : x → x + 3 and g : x → x².

(b)

$\frac{1}{\mathrm{x}}$ + $\frac{1}{\mathrm{x\; +\; 2}}$

For a function to be undefined, the denominator must be equal to 0.

$\frac{1}{\mathrm{x}}$ + $\frac{1}{\mathrm{x\; +\; 2}}$ = $\frac{1}{\mathrm{x}}$ + $\frac{1}{\mathrm{x\; +\; 2}}$

$\frac{1}{\mathrm{x}}$ + $\frac{1}{\mathrm{x\; +\; 2}}$ = $\frac{\mathrm{1\; x\; (x\; +\; 2)\; +\; x\; x\; 1}}{\mathrm{x(x\; +\; 2)}}$

$\frac{1}{\mathrm{x}}$ + $\frac{1}{\mathrm{x\; +\; 2}}$ = $\frac{\mathrm{x\; +\; 2\; +\; x}}{\mathrm{x(x\; +\; 2)}}$

$\frac{1}{\mathrm{x}}$ + $\frac{1}{\mathrm{x\; +\; 2}}$ = $\frac{\mathrm{2x\; +\; 2}}{\mathrm{x(x\; +\; 2)}}$

The denominator is x(x + 2).

The function is undefined when x = 0 or x + 2 = 0

When x + 2 = 0, x = -2

The function will be undefined when x = 0 or x = -2

(c)

f(x) = $\frac{\mathrm{k}}{\mathrm{x\; +\; 1}}$ + $\frac{6}{\mathrm{x\; +\; 2}}$

f(5) = $\frac{\mathrm{k}}{\mathrm{5\; +\; 1}}$ + $\frac{6}{\mathrm{5\; +\; 2}}$

f(5) = $\frac{\mathrm{k}}{6}$ + $\frac{6}{7}$

f(5) = $\frac{\mathrm{k\; x\; 7\; +\; 6\; x\; 6}}{42}$

f(5) = $\frac{\mathrm{7k\; +\; 36}}{42}$

But f(5) = 8

$\frac{\mathrm{7k\; +\; 36}}{42}$ = 8

Get rid of the fraction by multiplying both sides of the equation by the denominator 42

7k + 36 = 8 x 42

7k = 336 - 36

7k = 300

Divide both sides by 7

k = $\frac{300}{7}$ or 42$\frac{6}{7}$

(i)

f(-3) = 2 x (-3)² - 8 x(-3) + 5

f(-3) = 2 x 9 + 24 + 5

f(-3) = 18 + 24 + 5

f(-3) = 47

(ii)

f(x) = g(x)

2x² - 8x + 5 = x - 2

2x² - 8x + 5 - x + 2 = 0

2x² - 9x + 7 = 0

Notes:

1. a x c = 2 x 7 = 14

2. Split b(-9) such that two numbers when you multiply, you will get 14(ac) and when you add you will get b(-7)

3. The numbers are -7 and - 2. Hence -9x = -7x - 2x

2x² -7x - 2x + 7 = 0

x(2x - 7) - 1(2x - 7) = 0

(2x - 7)(x - 1) = 0

When 2x - 7 = 0, x = $\frac{7}{2}$

When x - 1 = 0, x = 1

∴ x = 1 or $\frac{7}{2}$