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Math  /  Algebra

QuestionFind the inverse of the function. f(x)=13x+103f−1(x)=\begin{array}{l} f(x)=\sqrt[3]{13 x+10} \\ f^{-1}(x)= \end{array} □\square
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Studdy Solution

STEP 1

1. We are given the function f(x)=13x+103 f(x) = \sqrt[3]{13x + 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) to express the function in terms of y y .
2. Solve the equation for x x in terms of y y .
3. Swap x x and y y to find the inverse function.
4. Write the inverse function f−1(x) f^{-1}(x) .

STEP 3

Set y=f(x) y = f(x) :
y=13x+103 y = \sqrt[3]{13x + 10}

STEP 4

To solve for x x , first cube both sides to eliminate the cube root:
y3=13x+10 y^3 = 13x + 10

STEP 5

Isolate x x by subtracting 10 from both sides:
y3−10=13x y^3 - 10 = 13x

STEP 6

Divide both sides by 13 to solve for x x :
x=y3−1013 x = \frac{y^3 - 10}{13}

STEP 7

Swap x x and y y to find the inverse function:
y=x3−1013 y = \frac{x^3 - 10}{13}

STEP 8

Write the inverse function f−1(x) f^{-1}(x) :
f−1(x)=x3−1013 f^{-1}(x) = \frac{x^3 - 10}{13}
The inverse function is:
f−1(x)=x3−1013 \boxed{f^{-1}(x) = \frac{x^3 - 10}{13}}

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