Use transformations of f(x)=1/x or f(x)=1/x2 to graph each rational function. g(x)=1/(x−1)

Ch. 3 - Polynomial and Rational Functions

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

All textbooks

Blitzer 8th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 45

Chapter 4, Problem 45

Use transformations of f(x)=1/x or f(x)=1/x² to graph each rational function. g(x)=1/(x−1)

Verified step by step guidance

Identify the base function given, which is $f(x) = \frac{1}{x}$. This is a rational function with a vertical asymptote at $x=0$ and a horizontal asymptote at $y=0$.

Recognize that the function $g(x) = \frac{1}{x-1}$ is a horizontal shift of the base function $f(x) = \frac{1}{x}$. The expression $x-1$ inside the denominator indicates a shift to the right by 1 unit.

Determine the new vertical asymptote by setting the denominator equal to zero: $x - 1 = 0$, which gives $x = 1$. This means the vertical asymptote moves from $x=0$ to $x=1$.

Note that the horizontal asymptote remains unchanged at $y=0$ because the degree of the numerator and denominator are the same and the leading coefficients are unchanged.

Sketch the graph by shifting the original graph of $f(x) = \frac{1}{x}$ one unit to the right, keeping the shape of the hyperbola the same but with the vertical asymptote at $x=1$ and the horizontal asymptote at $y=0$.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Parent Rational Functions

The parent functions f(x) = 1/x and f(x) = 1/x² are basic rational functions with distinct shapes and asymptotes. Understanding their graphs, including vertical and horizontal asymptotes, helps in visualizing transformations applied to these functions.

Transformations of Functions

Transformations involve shifting, stretching, compressing, or reflecting the graph of a function. For g(x) = 1/(x−1), the graph of f(x) = 1/x is shifted horizontally to the right by 1 unit, changing the location of asymptotes and key points.

Asymptotes of Rational Functions

Asymptotes are lines that the graph approaches but never touches. Vertical asymptotes occur where the denominator is zero, and horizontal asymptotes describe end behavior. Identifying these helps in accurately sketching the graph of rational functions.

Recommended video:

6:24

Introduction to Asymptotes

Related Practice

Textbook Question

Find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. f(x)=3x⁴−11x³−x²+19x+6

1073

views

Textbook Question

Give the domain and the range of each quadratic function whose graph is described. The vertex is and the parabola opens up.

1137

views

Textbook Question

In Exercises 45–46, describe in words the variation shown by the given equation. z = kx^2 √y

534

views

Textbook Question

Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. (−x+2)/(x−4)≥0

537

views

Textbook Question

In Exercises 39–52, find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. x⁴−3x³−20x²−24x−8=0

440

views

Textbook Question

Describe in words the variation shown by the given equation. $z = $\frac{k\sqrt{x}$}{y^2}$

612

views