Math / Algebra

QuestionFor the function $f(x)=\sqrt[5]{x}-2$ , find $f^{-1}(x)$ .

Studdy Solution

STEP 1

1. We are given the function $f(x) = \sqrt[5]{x} - 2$ .
2. We need to find the inverse function $f^{-1}(x)$ .

STEP 2

1. Set $y = f(x)$ and express it in terms of $x$ .
2. Solve for $x$ in terms of $y$ .
3. Swap $x$ and $y$ to find $f^{-1}(x)$ .
4. Verify the inverse function.

STEP 3

Set $y = f(x)$ :
$y = \sqrt[5]{x} - 2$

STEP 4

Solve for $x$ in terms of $y$ :
First, isolate the radical term:
$y + 2 = \sqrt[5]{x}$
Next, eliminate the fifth root by raising both sides to the power of 5:
$(y + 2)^5 = x$

STEP 5

Swap $x$ and $y$ to find the inverse function:
$f^{-1}(x) = (x + 2)^5$

STEP 6

Verify the inverse function by checking if $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$ .
First, check $f(f^{-1}(x))$ :
$f(f^{-1}(x)) = f((x + 2)^5) = \sqrt[5]{(x + 2)^5} - 2 = x + 2 - 2 = x$
Next, check $f^{-1}(f(x))$ :
$f^{-1}(f(x)) = f^{-1}(\sqrt[5]{x} - 2) = ((\sqrt[5]{x} - 2) + 2)^5 = (\sqrt[5]{x})^5 = x$
Both conditions are satisfied, confirming the inverse.
The inverse function is:
$f^{-1}(x) = (x + 2)^5$

Was this helpful?

QuestionFor the function $f(x)=\sqrt[5]{x}-2$ , find $f^{-1}(x)$ .

Studdy Solution

STEP 1

1. We are given the function $f(x) = \sqrt[5]{x} - 2$ .
2. We need to find the inverse function $f^{-1}(x)$ .

STEP 2

1. Set $y = f(x)$ and express it in terms of $x$ .
2. Solve for $x$ in terms of $y$ .
3. Swap $x$ and $y$ to find $f^{-1}(x)$ .
4. Verify the inverse function.

STEP 3

Set $y = f(x)$ :
$y = \sqrt[5]{x} - 2$

STEP 4

Solve for $x$ in terms of $y$ :
First, isolate the radical term:
$y + 2 = \sqrt[5]{x}$
Next, eliminate the fifth root by raising both sides to the power of 5:
$(y + 2)^5 = x$

STEP 5

Swap $x$ and $y$ to find the inverse function:
$f^{-1}(x) = (x + 2)^5$

STEP 6

Get Studdy

Studdy solves anything!

Start learning now

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

QuestionFor the function f(x)=x5−2f(x)=\sqrt[5]{x}-2f(x)=5x​−2, find f−1(x)f^{-1}(x)f−1(x).

Studdy Solution

STEP 1

1. We are given the function f(x)=x5−2 f(x) = \sqrt[5]{x} - 2 f(x)=5x​−2.2. We need to find the inverse function f−1(x) f^{-1}(x) f−1(x).

STEP 2

1. Set y=f(x) y = f(x) y=f(x) and express it in terms of x x x.2. Solve for x x x in terms of y y y.3. Swap x x x and y y y to find f−1(x) f^{-1}(x) f−1(x).4. Verify the inverse function.

STEP 3

Set y=f(x) y = f(x) y=f(x):y=x5−2 y = \sqrt[5]{x} - 2 y=5x​−2

STEP 4

Solve for x x x in terms of y y y:First, isolate the radical term:y+2=x5 y + 2 = \sqrt[5]{x} y+2=5x​Next, eliminate the fifth root by raising both sides to the power of 5:(y+2)5=x (y + 2)^5 = x (y+2)5=x

STEP 5

Swap x x x and y y y to find the inverse function:f−1(x)=(x+2)5 f^{-1}(x) = (x + 2)^5 f−1(x)=(x+2)5

STEP 6

Get Studdy

Studdy solves anything!

Start learning now

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

QuestionFor the function $f(x)=\sqrt[5]{x}-2$ , find $f^{-1}(x)$ .

1. We are given the function $f(x) = \sqrt[5]{x} - 2$ .
2. We need to find the inverse function $f^{-1}(x)$ .

1. Set $y = f(x)$ and express it in terms of $x$ .
2. Solve for $x$ in terms of $y$ .
3. Swap $x$ and $y$ to find $f^{-1}(x)$ .
4. Verify the inverse function.

Set $y = f(x)$ :
$y = \sqrt[5]{x} - 2$

Solve for $x$ in terms of $y$ :
First, isolate the radical term:
$y + 2 = \sqrt[5]{x}$
Next, eliminate the fifth root by raising both sides to the power of 5:
$(y + 2)^5 = x$

Swap $x$ and $y$ to find the inverse function:
$f^{-1}(x) = (x + 2)^5$