QuestionFind the inverse of $f(x)=\sqrt{x-5}$ for $x \geq 5$ .

Studdy Solution

STEP 1

Assumptions1. The function is $f(x)=\sqrt{x-5}$ . The domain of the function is $[5, \infty)$ 3. We need to find the inverse of the function

STEP 2

The first step in finding the inverse of a function is to replace the function notation $f(x)$ with $y$ .
$y = \sqrt{x-5}$

STEP 3

Next, we switch the roles of $x$ and $y$ . This means we replace every $x$ in our equation with $y$ and every $y$ with $x$ .
$x = \sqrt{y-5}$

STEP 4

Now, we need to solve this equation for $y$ to get the inverse function. First, we square both sides of the equation to eliminate the square root.
$(x)^2 = (\sqrt{y-})^2$

STEP 5

implify the equation.
$x^2 = y-5$

STEP 6

Add5 to both sides of the equation to isolate $y$ .
$x^2 +5 = y$

STEP 7

Now we have the inverse function. However, we need to consider the domain of the original function. The original function has a domain of $[5, \infty)$ , which means the range of the inverse function is $[5, \infty)$ . Therefore, we need to restrict the domain of the inverse function to $[0, \infty)$ to match the range of the original function.
The inverse function is $f^{-1}(x) = x^2 +5$ for $x \geq0$ .

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QuestionFind the inverse of $f(x)=\sqrt{x-5}$ for $x \geq 5$ .

Studdy Solution

STEP 1

Assumptions1. The function is $f(x)=\sqrt{x-5}$ . The domain of the function is $[5, \infty)$ 3. We need to find the inverse of the function

STEP 2

The first step in finding the inverse of a function is to replace the function notation $f(x)$ with $y$ .
$y = \sqrt{x-5}$

STEP 3

Next, we switch the roles of $x$ and $y$ . This means we replace every $x$ in our equation with $y$ and every $y$ with $x$ .
$x = \sqrt{y-5}$

STEP 4

Now, we need to solve this equation for $y$ to get the inverse function. First, we square both sides of the equation to eliminate the square root.
$(x)^2 = (\sqrt{y-})^2$

STEP 5

implify the equation.
$x^2 = y-5$

STEP 6

Add5 to both sides of the equation to isolate $y$ .
$x^2 +5 = y$

STEP 7

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QuestionFind the inverse of f(x)=x−5f(x)=\sqrt{x-5}f(x)=x−5​ for x≥5x \geq 5x≥5.

Studdy Solution

STEP 1

Assumptions1. The function is f(x)=x−5f(x)=\sqrt{x-5}f(x)=x−5​ . The domain of the function is [5,∞)[5, \infty)[5,∞)3. We need to find the inverse of the function

STEP 2

The first step in finding the inverse of a function is to replace the function notation f(x)f(x)f(x) with yyy.y=x−5y = \sqrt{x-5}y=x−5​

STEP 3

Next, we switch the roles of xxx and yyy. This means we replace every xxx in our equation with yyy and every yyy with xxx.x=y−5x = \sqrt{y-5}x=y−5​

STEP 4

Now, we need to solve this equation for yyy to get the inverse function. First, we square both sides of the equation to eliminate the square root.(x)2=(y−)2(x)^2 = (\sqrt{y-})^2(x)2=(y−​)2

STEP 5

implify the equation.x2=y−5x^2 = y-5x2=y−5

STEP 6

Add5 to both sides of the equation to isolate yyy.x2+5=yx^2 +5 = yx2+5=y

STEP 7

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QuestionFind the inverse of $f(x)=\sqrt{x-5}$ for $x \geq 5$ .

Assumptions1. The function is $f(x)=\sqrt{x-5}$ . The domain of the function is $[5, \infty)$ 3. We need to find the inverse of the function

The first step in finding the inverse of a function is to replace the function notation $f(x)$ with $y$ .
$y = \sqrt{x-5}$

Next, we switch the roles of $x$ and $y$ . This means we replace every $x$ in our equation with $y$ and every $y$ with $x$ .
$x = \sqrt{y-5}$

Now, we need to solve this equation for $y$ to get the inverse function. First, we square both sides of the equation to eliminate the square root.
$(x)^2 = (\sqrt{y-})^2$

implify the equation.
$x^2 = y-5$

Add5 to both sides of the equation to isolate $y$ .
$x^2 +5 = y$