Chapter 2, Problem 10RE

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

$f\left(x\right)\text{ = }\frac{\sqrt{x+1}}{x^2-4}$

To determine

To find: The domain of the provided function.

Answer to Problem 10RE

Solution:

The domain of the function $f$ is $\left\{x\right|x\ne 2,x\ne -2\}$.

Explanation of Solution

Given:

It is provided in the problem that the function is $f\left(x\right)=\frac{\sqrt{x+1}}{x^2-4}$.

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 $\sqrt{x+1}$ by $x^2-4$.

Let’s factorize the denominator,

$x^2-4=x^2-2^2$

$=\left(x-2\right)\left(x+2\right)$

Since division by 0 is not defined, the denominator $\left(x-2\right)\left(x+2\right)$ can never be 0, so $x$ can never equal 2 or $-2$. The domain of the function $f$ is $\left\{x\right|x\ne 2,x\ne -2\}$.