QuestionFind the function $\left(\frac{f}{g}\right)(x)$ where $f(x)=x^{2}-3x+2$ and $g(x)=x-2$ . What is the domain?

Studdy Solution

STEP 1

Assumptions1. The function $f(x)$ is defined as $f(x) = x^{} -3x +$ . The function $g(x)$ is defined as $g(x) = x -$
3. We are asked to find the function $\left(\frac{f}{g}\right)(x)$ and its domain

STEP 2

First, we need to find the function $\left(\frac{f}{g}\right)(x)$ . This is done by dividing the function $f(x)$ by the function $g(x)$ .
$\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$

STEP 3

Now, plug in the given functions $f(x)$ and $g(x)$ into the equation.
$\left(\frac{f}{g}\right)(x) = \frac{x^{2} -3x +2}{x -2}$

STEP 4

The function $\left(\frac{f}{g}\right)(x)$ is now defined as $\left(\frac{f}{g}\right)(x) = \frac{x^{2} -3x +2}{x -2}$

STEP 5

Next, we need to find the domain of $\left(\frac{f}{g}\right)(x)$ . The domain of a function is the set of all possible input values (x-values) which will output real numbers.For a function of the form $\frac{f(x)}{g(x)}$ , the domain is all real numbers except for those that make $g(x) =0$ , because division by zero is undefined.

STEP 6

To find the values that make $g(x) =0$ , we set $g(x)$ equal to zero and solve for $x$ .
$x -2 =0$

STEP 7

olving the equation $x -2 =0$ gives us $x =2$ .

STEP 8

Therefore, the domain of $\left(\frac{f}{g}\right)(x)$ is all real numbers except $x =2$ .
In mathematical notation, this can be written as $x \neq2$ .
So, $\left(\frac{f}{g}\right)(x) = \frac{x^{2} -3x +2}{x -2}$ and the domain of $\left(\frac{f}{g}\right)(x)$ is $x \neq2$ .

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