Question Find the expression of $\left(\frac{f}{g}\right)(x)$ , where $f(x)=x^{2}-16x+64$ and $g(x)=x-8$ , and express the answer in interval notation.

Studdy Solution

STEP 1

Assumptions1. We have two functions $f(x) = x^{}-16x+64$ and $g(x) = x-8$ . . We are asked to find the function $\left(\frac{f}{g}\right)(x)$ .
3. We are asked to express the answer in interval notation.

STEP 2

First, let's 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.
$\left(\frac{f}{g}\right)(x) = \frac{x^{2}-16x+64}{x-8}$

STEP 4

Now, we need to simplify the fraction. We can do this by factoring the numerator.
$x^{2}-16x+64 = (x-8)^2$ So, the fraction becomes $\left(\frac{f}{g}\right)(x) = \frac{(x-8)^2}{x-8}$

STEP 5

Next, we simplify the fraction by cancelling out the common factor of $x-8$ in the numerator and denominator.
$\left(\frac{f}{g}\right)(x) = x-8$

STEP 6

Now, we need to find the domain of $\left(\frac{f}{g}\right)(x)$ . The domain is all real numbers except for the values of $x$ that make the denominator of the original fraction equal to zero.
$x-8 =0$ olving for $x$ , we get $x =8$ .

STEP 7

So, the domain of $\left(\frac{f}{g}\right)(x)$ is all real numbers except $x =$ . In interval notation, this is expressed as $(-\infty,) \cup (, \infty)$ So, $\left(\frac{f}{g}\right)(x) = x-$ for $x \in (-\infty,) \cup (, \infty)$ .

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Question Find the expression of $\left(\frac{f}{g}\right)(x)$ , where $f(x)=x^{2}-16x+64$ and $g(x)=x-8$ , and express the answer in interval notation.

Studdy Solution

STEP 1

Assumptions1. We have two functions $f(x) = x^{}-16x+64$ and $g(x) = x-8$ . . We are asked to find the function $\left(\frac{f}{g}\right)(x)$ .
3. We are asked to express the answer in interval notation.

STEP 2

First, let's 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.
$\left(\frac{f}{g}\right)(x) = \frac{x^{2}-16x+64}{x-8}$

STEP 4

Now, we need to simplify the fraction. We can do this by factoring the numerator.
$x^{2}-16x+64 = (x-8)^2$ So, the fraction becomes $\left(\frac{f}{g}\right)(x) = \frac{(x-8)^2}{x-8}$

STEP 5

Next, we simplify the fraction by cancelling out the common factor of $x-8$ in the numerator and denominator.
$\left(\frac{f}{g}\right)(x) = x-8$

STEP 6

Now, we need to find the domain of $\left(\frac{f}{g}\right)(x)$ . The domain is all real numbers except for the values of $x$ that make the denominator of the original fraction equal to zero.
$x-8 =0$ olving for $x$ , we get $x =8$ .

STEP 7

So, the domain of $\left(\frac{f}{g}\right)(x)$ is all real numbers except $x =$ . In interval notation, this is expressed as $(-\infty,) \cup (, \infty)$ So, $\left(\frac{f}{g}\right)(x) = x-$ for $x \in (-\infty,) \cup (, \infty)$ .

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Question Find the expression of (fg)(x)\left(\frac{f}{g}\right)(x)(gf​)(x), where f(x)=x2−16x+64f(x)=x^{2}-16x+64f(x)=x2−16x+64 and g(x)=x−8g(x)=x-8g(x)=x−8, and express the answer in interval notation.

Studdy Solution

STEP 1

Assumptions1. We have two functions f(x)=x−16x+64f(x) = x^{}-16x+64f(x)=x−16x+64 and g(x)=x−8g(x) = x-8g(x)=x−8. . We are asked to find the function (fg)(x)\left(\frac{f}{g}\right)(x)(gf​)(x).3. We are asked to express the answer in interval notation.

STEP 2

First, let's find the function (fg)(x)\left(\frac{f}{g}\right)(x)(gf​)(x). This is done by dividing the function f(x)f(x)f(x) by the function g(x)g(x)g(x).(fg)(x)=f(x)g(x)\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}(gf​)(x)=g(x)f(x)​

STEP 3

Now, plug in the given functions f(x)f(x)f(x) and g(x)g(x)g(x) into the equation.(fg)(x)=x2−16x+64x−8\left(\frac{f}{g}\right)(x) = \frac{x^{2}-16x+64}{x-8}(gf​)(x)=x−8x2−16x+64​

STEP 4

Now, we need to simplify the fraction. We can do this by factoring the numerator.x2−16x+64=(x−8)2x^{2}-16x+64 = (x-8)^2x2−16x+64=(x−8)2So, the fraction becomes(fg)(x)=(x−8)2x−8\left(\frac{f}{g}\right)(x) = \frac{(x-8)^2}{x-8}(gf​)(x)=x−8(x−8)2​

STEP 5

Next, we simplify the fraction by cancelling out the common factor of x−8x-8x−8 in the numerator and denominator.(fg)(x)=x−8\left(\frac{f}{g}\right)(x) = x-8(gf​)(x)=x−8

STEP 6

Now, we need to find the domain of (fg)(x)\left(\frac{f}{g}\right)(x)(gf​)(x). The domain is all real numbers except for the values of xxx that make the denominator of the original fraction equal to zero.x−8=0x-8 =0x−8=0olving for xxx, we get x=8x =8x=8.

STEP 7

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Question Find the expression of $\left(\frac{f}{g}\right)(x)$ , where $f(x)=x^{2}-16x+64$ and $g(x)=x-8$ , and express the answer in interval notation.

Assumptions1. We have two functions $f(x) = x^{}-16x+64$ and $g(x) = x-8$ . . We are asked to find the function $\left(\frac{f}{g}\right)(x)$ .
3. We are asked to express the answer in interval notation.

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

Now, plug in the given functions $f(x)$ and $g(x)$ into the equation.
$\left(\frac{f}{g}\right)(x) = \frac{x^{2}-16x+64}{x-8}$

Now, we need to simplify the fraction. We can do this by factoring the numerator.
$x^{2}-16x+64 = (x-8)^2$ So, the fraction becomes $\left(\frac{f}{g}\right)(x) = \frac{(x-8)^2}{x-8}$

Next, we simplify the fraction by cancelling out the common factor of $x-8$ in the numerator and denominator.
$\left(\frac{f}{g}\right)(x) = x-8$

Now, we need to find the domain of $\left(\frac{f}{g}\right)(x)$ . The domain is all real numbers except for the values of $x$ that make the denominator of the original fraction equal to zero.
$x-8 =0$ olving for $x$ , we get $x =8$ .