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Analyze the function f(x)= (x^3-4x^2-31x+70)/(x^2-5x+6). Find the y-intercept, the...

Question:

Analyze the function

{eq}f(x)= \frac{x^3 - 4x^2 - 31x + 70}{x^2 - 5x + 6} {/eq}.

Find the y-intercept, the x-intercept(s), the removable singularities, the vertical asymptotes and the horizontal asymptotes.

1) How many x-intercepts does {eq}f {/eq} have?

2) How many removable singularities does {eq}f {/eq} have?

3) How many vertical asymptotes does {eq}f {/eq} have?

4) How many horizontal asymptotes does {eq}f {/eq} have?

What is the y-intercept of the graph {eq}y=f(x) {/eq}?

What are the x-intercepts of the graph {eq}y=f(x) {/eq}?

What is the x-value of the removable singularity?

What is the limit of {eq}f(x) {/eq} as {eq}x {/eq} approaches this removable singularity?

What is the equation of the vertical asymptote?

What is the limit of {eq}f(x) {/eq} as {eq}x {/eq} approaches this vertical asymptote from the positive side?

What is the limit of {eq}f(x) {/eq} as {eq}x {/eq} approaches this vertical asymptote from the negative side?

Rational Functions

Rational functions are functions which are ratios of polynomials. Rational functions often have vertical asymptotes, horizontal asymptotes, holes (or removable singularities), and slant asymptotes. To determine if a rational function has a vertical asymptote or a hole, the numerator and denominator should be fully factored. Any factors which cancel completely out of the denominator correspond to a removable singularity (or hole) in the graph. Any factors which are still in the denominator correspond to a vertical asymptote. Horizontal asymptotes can be found by comparing the degrees of the numerator and denominator. A higher degree in the numerator results in no horizontal asymptotes, a higher degree in the denominator results in a horizontal asymptote at the x axis, and equal degrees results in a horizontal asymptote at the ratio of the leading coefficients.

Answer and Explanation:

1) How many x-intercepts does {eq}f {/eq} have?


First notice that the domain of the function is {eq}(\infty, 2)\cup(2,3)\cup(3,\infty) {/eq}. To find x intercepts, set the function equal to zero.

{eq}\frac{x^3 - 4x^2 - 31x + 70}{x^2 - 5x + 6} = 0\\ x^3 - 4x^2 - 31x + 70=0\\ (x-7)(x-2)(x+5)=0\\ x=7,2,-5 {/eq}
But since 2 is not in the domain, we only have two x intercepts {eq}(7,0) , (-5,0) {/eq}


2) How many removable singularities does {eq}f {/eq} have?


The numerator and denominator have one factor in common, and so there is one removable singularity, or hole, at {eq}(2, 35) {/eq}


3) How many vertical asymptotes does {eq}f {/eq} have?


There is one vertical asymptote corresponding to the factor {eq}x-3 {/eq} in the denominator. The vertical asymptote is {eq}x=3 {/eq}


4) How many horizontal asymptotes does {eq}f {/eq} have?


There are no horizontal asymptotes, since the degree of the numerator is larger than the degree of the denominator


What is the y-intercept of the graph {eq}y=f(x) {/eq}?


To find a y intercept, substitute x=0. The y intercept is {eq}(0,\frac{35}{3}) {/eq}


What are the x-intercepts of the graph {eq}y=f(x) {/eq}?


The x intercepts are {eq}(7,0) , (-5,0) {/eq}


What is the x-value of the removable singularity?


The x value of the removable singularity is x=2


What is the limit of {eq}f(x) {/eq} as {eq}x {/eq} approaches this removable singularity?


As x approaches the removable singularity, the function approaches 35.


What is the equation of the vertical asymptote?


{eq}x=3 {/eq}


What is the limit of {eq}f(x) {/eq} as {eq}x {/eq} approaches this vertical asymptote from the positive side?


The limit is {eq}-\infty {/eq}, as the denominator approaches 0 but is positive and the numerator is negative


What is the limit of {eq}f(x) {/eq} as {eq}x {/eq} approaches this vertical asymptote from the negative side?


The limit is {eq}\infty {/eq}, since the denominator approaches 0 but is negative and the numerator is negative


Learn more about this topic:

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Rational Function: Definition, Equation & Examples

from GMAT Prep: Help and Review

Chapter 10 / Lesson 11
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