Skip to main content
Ch. 3 - Polynomial and Rational Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 3 - Polynomial and Rational FunctionsProblem 2
Chapter 4, Problem 2

Provide a short answer to each question. What is the domain of the function ƒ(x)=1x2ƒ(x)=\(\frac{1}{x^2}\)? What is its range?

Verified step by step guidance
1
Identify the domain by determining all values of \(x\) for which the function \(f(x) = \frac{1}{x^2}\) is defined. Since division by zero is undefined, exclude values that make the denominator zero.
Set the denominator equal to zero and solve: \(x^2 = 0\). This gives \(x = 0\), which must be excluded from the domain.
Conclude that the domain is all real numbers except \(x = 0\), which can be written as \((-\infty, 0) \cup (0, \infty)\).
To find the range, analyze the output values of \(f(x) = \frac{1}{x^2}\). Since \(x^2\) is always positive for \(x \neq 0\), \(\frac{1}{x^2}\) is also always positive.
Note that as \(x\) approaches zero, \(f(x)\) grows without bound, and as \(x\) approaches infinity or negative infinity, \(f(x)\) approaches zero but never reaches it. Therefore, the range is \((0, \infty)\).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
5m
Was this helpful?

Key Concepts

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

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For rational functions like ƒ(x) = 1/x², the domain excludes values that make the denominator zero, as division by zero is undefined.
Recommended video:
3:51
Domain Restrictions of Composed Functions

Range of a Function

The range of a function is the set of all possible output values (y-values) the function can produce. For ƒ(x) = 1/x², since x² is always positive except at zero, the function outputs positive values, and the range reflects these possible outputs.
Recommended video:
4:22
Domain & Range of Transformed Functions

Behavior of Rational Functions with Squared Denominators

In functions like ƒ(x) = 1/x², the denominator is squared, making it always positive except at zero. This affects the function's behavior by ensuring outputs are positive and the function approaches infinity near zero, influencing both domain restrictions and range values.
Recommended video:
Guided course
02:58
Rationalizing Denominators
Related Practice
Textbook Question

Graph each quadratic function. Give the vertex, axis, x-intercepts, y-intercept, domain, range, and largest open intervals of the domain over which each function is increasing or decreasing. ƒ(x)=-3x2-12x-1

4
views
Textbook Question

Fill in the blank(s) to correctly complete each sentence. The highest point on the graph of a parabola that opens down is the ____ of the parabola.

3
views
Textbook Question

Fill in the blank(s) to correctly complete each sentence. A polynomial function with leading term 3x5 has degree ____.

Textbook Question

Determine whether each statement is true or false. If false, explain why. Because x-1 is a factor of ƒ(x)=x6-x4+2x2-2, we can also conclude that ƒ(1) = 0.

1
views
Textbook Question

Provide a short answer to each question. What is the domain of the function ƒ(x)=1/x? What is its range?

7
views
Textbook Question

Use the graph to solve each equation or inequality. Use interval notation where appropriate. 7x(x - 1)(x - 2) = 0