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Chapter 2: Problem 3

Find the domain of each rational function. $$g(x)=\frac{3 x^{2}}{(x-5)(x+4)}$$

Short Answer

Expert verified
The domain of the function \( g(x)=\frac{3 x^{2}}{(x-5)(x+4)} \) is all real numbers except \( x=5 \) and \( x=-4 \).

Step by step solution

01

Identify the denominator

In the function \( g(x)=\frac{3 x^{2}}{(x-5)(x+4)} \), the denominator is \( (x-5)(x+4) \). This is what we will set equal to zero to find the values where \( g(x) \) is undefined.
02

Set the denominator equal to zero

Setting the denominator equal to zero gives us the equation \( (x-5)(x+4) = 0 \). This is a simple quadratic equation, which can be solved by applying the zero-product property, which says that if a product of factors is zero, then at least one of the factors must be zero.
03

Solve for x

Setting each factor equal to zero gives us the equations \( x-5=0 \) and \( x+4=0 \). Solving these for \( x \) gives \( x=5 \) and \( x=-4 \), respectively.
04

Determine the domain

The domain of \( g(x) \) is all real numbers except for \( x=5 \) and \( x=-4 \). Thus, the domain of \( g(x) \) is \( x \in R \backslash \{5, -4\} \).

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Most popular questions from this chapter

A tourist drives 90 miles along a scenic highway and then takes a 5-mile walk along a hiking trail. The average velocity driving is nine times that while hiking. Express the total time for driving and hiking, \(T,\) as a function of the average velocity on the hike, \(x\).

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