Graph each rational function. ƒ(x)=5x/(x2-1)

Ch. 3 - Polynomial and Rational Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

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Lial 13th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 69

Chapter 4, Problem 69

Graph each rational function. ƒ(x)=5x/(x²-1)

Verified step by step guidance

Identify the rational function given: $f(x) = \frac{5x}{x^{2} - 1}$. Notice that the denominator can be factored.

Factor the denominator: $x^{2} - 1 = (x - 1)(x + 1)$. This helps us find the domain restrictions and vertical asymptotes.

Determine the domain by setting the denominator not equal to zero: solve $x^{2} - 1 \neq 0$, which means $x \neq 1$ and $x \neq -1$. These values are vertical asymptotes.

Find the horizontal asymptote by comparing the degrees of the numerator and denominator. Since the degree of the denominator (2) is greater than the numerator (1), the horizontal asymptote is $y = 0$.

Find the x-intercept by setting the numerator equal to zero: $5x = 0$, so $x = 0$. Find the y-intercept by evaluating $f(0)$, which is $\frac{0}{0^{2} - 1} = 0$. Use these points and asymptotes to sketch the graph.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x), where Q(x) ≠ 0. Understanding the domain, zeros, and behavior of these functions is essential for graphing, as they often have asymptotes and discontinuities where the denominator is zero.

Vertical and Horizontal Asymptotes

Vertical asymptotes occur where the denominator equals zero and the function is undefined, indicating values the graph approaches but never touches. Horizontal asymptotes describe the end behavior of the function as x approaches infinity or negative infinity, determined by comparing the degrees of numerator and denominator polynomials.

Graphing Techniques for Rational Functions

Graphing involves finding intercepts, asymptotes, and analyzing the function's behavior near discontinuities. Plotting key points and understanding symmetry or sign changes help create an accurate sketch of the function's curve.

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Related Practice

Textbook Question

Solve each rational inequality. Give the solution set in interval notation.

$\frac{5}{2 - x} > \frac{3}{3 - x}$

Textbook Question

Find a polynomial function f of least degree having the graph shown. (Hint: See the NOTE following Example 4.)

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Textbook Question

Solve each rational inequality. Give the solution set in interval notation. 1 /{x² - 4x + 3} ≤ 1 /{ 3 - x}

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Textbook Question

The remainder theorem indicates that when a polynomial ƒ(x) is divided by x-k, the remainder is equal to ƒ(k). Consider the polynomial function ƒ(x) = x³ - 2x² - x+2. Use the remainder theorem to find each of the following. Then determine the coordinates of the corresponding point on the graph of ƒ(x). ƒ (1)

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Textbook Question

Graph each rational function. See Examples 5–9.

$ƒ (x) = [(x + 6) (x - 2)] / [(x + 3) (x - 4)]$

Textbook Question

Graph each rational function. See Examples 5–9.

$ƒ (x) = x / (4 - x^{2})$

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