2 of 232 of 23 Items

Question

Question
Find the domain of the function left-parenthesis Start Fraction f over g End Fraction right-parenthesis left-parenthesis x right-parenthesis comma given f left-parenthesis x right-parenthesis equals x squared minus 3 x minus 10 and g left-parenthesis x right-parenthesis equals x squared minus 4 .
(1 point)
Responses

left-brace x vertical line x not equal to 2 right-brace
Image with alt text: left-brace x vertical line x not equal to 2 right-brace

left-brace x vertical line x not equal to negative 2 or 5 right-brace
Image with alt text: left-brace x vertical line x not equal to negative 2 or 5 right-brace

left-brace x vertical line x not equal to negative 2 or 2 right-brace
Image with alt text: left-brace x vertical line x not equal to negative 2 or 2 right-brace

left-brace x vertical line x not equal to 5 right-brace
Image with alt text: left-brace x vertical line x not equal to 5 right-brace

Answer 1

To find the domain of the function $\frac{f}{g}(x)$, where $f(x)=x^2-3x-10$ and $g(x)=x^2-4$, we need to determine when $g(x)\ne 0$ because a function is undefined when its denominator is zero.

First, let's solve for when $g(x)=0$:

$$g(x)=x^2-4=0$$

Factoring the quadratic, we find:

$$(x-2)(x+2)=0$$

Setting each factor to zero gives us:

$$x-2=0\Rightarrow x=2$$ $$x+2=0\Rightarrow x=-2$$

Thus, $g(x)=0$ when $x=2$ or $x=-2$. Therefore, the function $\frac{f}{g}(x)$ is undefined at these points.

The domain of the function $\frac{f}{g}(x)$ is all real numbers except for these points. In interval notation, this is:

$$\text{Domain}=x,|,x\ne -2\text{ and }x\ne 2$$

Thus, the correct response is:

left-brace x vertical line x not equal to negative 2 or 2 right-brace