/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 7 Find the domain of each function... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 7

Find the domain of each function. $$g(x)=\frac{3}{x^{2}-2 x-15}$$

Short Answer

Expert verified
The domain of the function \(g(x) = \frac{3}{{x^{2}-2x-15}}\) is all real numbers, except \(x \neq 5\) and \(x \neq -3\).

Step by step solution

01

Identify the Function Type

The provided function \(g(x) = \frac{3}{{x^{2}-2x-15}}\) is a rational function where the numerator and denominator are polynomials. The denominator cannot be zero because division by zero is undefined in mathematics.
02

Find Values that Makes the Denominator Zero

To determine the restrictions on the domain, we need to identify the values that make the denominator zero. Therefore, we solve the equation \(0 = x^{2} - 2x - 15\). This equation can be factored to become \(0 = (x-5)(x+3)\). Thus, \(x = 5\) and \(x = -3\) make the denominator zero.
03

Define the Domain

The domain of a function is the set of all values of \(x\) for which the function is defined. For this function, it is defined for all real numbers except for the points where the denominator is zero. So, the domain of \(g(x)\) is \(x \in \mathbb{R}\), except \(x \neq 5\) and \(x \neq -3\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

I noticed that the difference quotient is always zero if \(f(x)=c,\) where \(c\) is any constant.

In your own words, describe how to find the distance between two points in the rectangular coordinate system.

What is a circle? Without using variables, describe how the definition of a circle can be used to obtain a form of its equation.

Use a graphing utility to graph each circle whose equation is given. Use a square setting for the viewing window. $$x^{2}+10 x+y^{2}-4 y-20=0$$

Graph both equations in the same rectangular coordinate system and find all points of intersection. Then show that these ordered pairs satisfy the equations. $$\begin{aligned}(x-3)^{2}+(y+1)^{2} &=9 \\\y &=x-1\end{aligned}$$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.