Chapter 2, Problem 9RE

In Problems 6-11, find the domain of each function.

$f\left(x\right)\text{ = }\frac{x}{x^2+2x-3}$

To determine

To find: The domain of the provided function.

Answer to Problem 9RE

Solution:

The domain of the function $f$ is $\left\{x\right|x\ne 3,x\ne -1\}$.

Explanation of Solution

Given:

It is provided in the problem that the function is $f\left(x\right)=\frac{x}{x^2+2x-3}$.

Formula used:

Rules used to find the domain of a function defined by an equation:

1. Start with the domain as the set of all numbers.

2. If the equation has a denominator, exclude any numbers that give a zero denominator.

3. If the equation has a radical of even index, exclude any numbers that cause the expression inside the radical (the radicand) to be negative.

The function $f$ says to divide $x$ by $x^2+2x-3$.

Let’s factorize the denominator,

$x^2+2x-3=x^2-3x+x-3$

$=x\left(x-3\right)+1\left(x-3\right)$

$=\left(x-3\right)\left(x+1\right)$

Since division by 0 is not defined, the denominator $\left(x-3\right)\left(x+1\right)$ can never be 0, so $x$ can never equal 3 or $-1$. The domain of the function $f$ is $\left\{x\right|x\ne 3,x\ne -1\}$.