Let's begin by copying the function.

g (x) = \frac{x - a}{x - b}

We will first find the intercepts of the function.

Intercepts

To find the

x -

intercept we substitute

g (x)

and solve the resulting equation for

x .

g (x) = \frac{x - a}{x - b}

Substitute

$g (x) = 0$

0 = \frac{x - a}{x - b}

MultEqn

$LHS \cdot (x - b) = RHS \cdot (x - b)$

0 = x - a

AddEqn

$LHS + a = RHS + a$

a = x

RearrangeEqn

Rearrange equation

x = a

The

x -

intercept is the point

(a, 0) .

To find the

y -

intercept of it, we substitute

x = 0

in the function and solve the resulting equation for

g (x) .

g (x) = \frac{x - a}{x - b}

Substitute

$x = 0$

g (0) = \frac{0 - a}{0 - b}

SubTerms

Subtract terms

g (0) = \frac{- a}{- b}

DivNegNeg

$\frac{- a}{- b} = \frac{a}{b}$

g (0) = \frac{a}{b}

The

y -

intercept is the point

(0, \frac{a}{b}) .

Next we will determine the asymptotes.

Asymptotes

The vertical asymptotes of a rational function are the

x -

values that make the denominator

0 .

To find the vertical asymptotes, we set the denominator equal to

0 .

x - b = 0 \Rightarrow x = b

Therefore, the vertical asymptote of the function is a vertical line where

x = b .

To find the horizontal asymptote, we should examine the degrees and the leading coefficients of the polynomials in the numerator and denominator.

g (x) = \frac{1 x ^{1} - a}{1 x ^{1} - b}

The degrees of the polynomials are the same. Therefore, the horizontal asymptote is the line where

y

equals the ratio of the leading coefficients.

y = 1

Knowing the asymptotes, we can next find the domain and range of the function.

Domain and Range

To determine the domain and range, we exclude the vertical asymptote from the domain and we exclude the horizontal asymptote from the range. Therefore, they can be determined as below.

Domain : Range : All real numbers except b All real numbers except 1

Exercise