QuestionFind the domain of $(s \cdot t)(x)=\frac{-1}{x+6}$ , where $s(x)=\frac{x-3}{x^{2}-36}$ and $t(x)=\frac{x-6}{3-x}$ .

Studdy Solution

STEP 1

Assumptions1. The product function is $(s \cdot t)(x)=\frac{-1}{x+6}$ , where $s(x)=\frac{x-3}{x^{}-36}$ and $t(x)=\frac{x-6}{3-x}$ .. The domain of a function is the set of all possible input values (x-values) which will output real numbers.

STEP 2

First, we need to find the domain of each function $s(x)$ and $t(x)$ . The domain of a function is the set of all real numbers for which the function is defined. For a rational function, the function is undefined when the denominator is equal to zero.
For $s(x)$ , we set the denominator equal to zero and solve for $x$ .
$x^{2}-36=0$

STEP 3

olve the equation $x^{2}-36=0$ for $x$ .
$x^{2}=36$ $x=\pm\sqrt{36}$ $x=\pm6$

STEP 4

So, the function $s(x)$ is undefined at $x=6$ and $x=-6$ . Therefore, the domain of $s(x)$ is all real numbers except $6$ and $-6$ .
Now, let's find the domain of $t(x)$ . Set the denominator equal to zero and solve for $x$ .
$3-x=0$

STEP 5

olve the equation $3-x=0$ for $x$ .
$x=3$

STEP 6

So, the function $t(x)$ is undefined at $x=3$ . Therefore, the domain of $t(x)$ is all real numbers except $3$ .

STEP 7

Now, let's find the domain of the product function $(s \cdot t)(x)$ . Set the denominator equal to zero and solve for $x$ .
$x+6=0$

STEP 8

olve the equation $x+6=0$ for $x$ .
$x=-6$

STEP 9

So, the function $(s \cdot t)(x)$ is undefined at $x=-6$ . Therefore, the domain of $(s \cdot t)(x)$ is all real numbers except $-6$ .

STEP 10

Finally, to find the domain of the product function $(s \cdot t)(x)$ , we need to consider the domains of $s(x)$ , $t(x)$ , and $(s \cdot t)(x)$ . The domain of the product function is the intersection of the domains of $s(x)$ , $t(x)$ , and $(s \cdot t)(x)$ .
The domain of the product function $(s \cdot t)(x)$ is all real numbers except $-6$ , $6$ , and $3$ .

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms