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Math  /  Algebra

QuestionFor the function f(x)=10(x+8)7−10f(x)=10(x+8)^{7}-10, find f−1(x)f^{-1}(x).

Studdy Solution

STEP 1

1. We are given the function f(x)=10(x+8)7−10 f(x) = 10(x+8)^7 - 10 .
2. We need to find the inverse function f−1(x) f^{-1}(x) .

STEP 2

1. Set y=f(x) y = f(x) .
2. Solve the equation for x x in terms of y y .
3. Express the inverse function f−1(x) f^{-1}(x) using the solved equation.

STEP 3

Set y=f(x) y = f(x) :
y=10(x+8)7−10 y = 10(x+8)^7 - 10

STEP 4

Add 10 to both sides of the equation to isolate the term with x x :
y+10=10(x+8)7 y + 10 = 10(x+8)^7

STEP 5

Divide both sides by 10 to further isolate the term with x x :
y+1010=(x+8)7 \frac{y + 10}{10} = (x+8)^7

STEP 6

Take the seventh root of both sides to solve for x+8 x + 8 :
y+10107=x+8 \sqrt[7]{\frac{y + 10}{10}} = x + 8

STEP 7

Subtract 8 from both sides to solve for x x :
x=y+10107−8 x = \sqrt[7]{\frac{y + 10}{10}} - 8

STEP 8

Express the inverse function f−1(x) f^{-1}(x) by replacing y y with x x :
f−1(x)=x+10107−8 f^{-1}(x) = \sqrt[7]{\frac{x + 10}{10}} - 8
The inverse function is:
f−1(x)=x+10107−8 f^{-1}(x) = \sqrt[7]{\frac{x + 10}{10}} - 8

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