Chapter 11, Problem 39CR

To determine

Find the asymptotes of the function $f(x)=\frac{3}{x-5}-1$ and then graph it.

Answer to Problem 39CR

Vertical asymptote: x = 5 and Horizontal Asymptote y = − 1

Explanation of Solution

Given:

Find the asymptotes of the function $f(x)=\frac{3}{x-5}-1$ and then graph it.

Concept Used:

To find the vertical asymptote(s) of a rational function, simply set the denominator equal to 0 and solve for x. We must set the denominator equal to 0 and solve.

The vertical asymptotes will occur at those values of x for which the denominator is equal to zero.

To find horizontal asymptotes: If the degree (the largest exponent) of the denominator is bigger than the degree of the numerator, the horizontal asymptote is the x-axis (y = 0).

If the degree of the numerator is bigger than the denominator, there is no horizontal asymptote

The degree of numerator is equal to the degree of the denominator. In this case the asymptote is the horizontal line $y=\frac{a}{b}$ , where the leading coefficient in the numerator is $a$ and $b$ is the leading coefficient in the denominator.

Calculation:

The vertical asymptotes will occur at those values of x for which the denominator is equal to zero.

Vertical asymptote: $x-5\ne 0\Rightarrow x\ne 5$

Simplified function: $f(x)=\frac{3}{x-5}-1\Rightarrow f(x)=\frac{3}{x-5}-\frac{x-5}{x-5}\Rightarrow f(x)=\frac{3-x+5}{x-5}\Rightarrow f(x)=\frac{-x+8}{x-5}$

The degree of numerator is equal to the degree of the denominator. In this case the asymptote is the horizontal line $y=\frac{a}{b}$ , where the leading coefficient in the numerator is $a$ and $b$ is the leading coefficient in the denominator. Here $a=-1;b=1;\text{H}\text{A}:y=\frac{a}{b}=-1$

Horizontal Asymptote y = − 1

Vertical asymptote: $x-5\ne 0\Rightarrow x\ne 5$ and Horizontal Asymptote y = − 1

The graph of the function: $f(x)=\frac{3}{x-5}-1$

  

Thus, Vertical asymptote: x = 5 and Horizontal Asymptote y = − 1