Math

QuestionFind the inverse of the one-to-one function f(x)=x38f(x)=x^{3}-8.

Studdy Solution

STEP 1

Assumptions1. The function is one-to-one, which means it has an inverse. . The function is f(x)=x38f(x) = x^{3} -8.

STEP 2

To find the inverse of a function, we first replace the function notation f(x)f(x) with yy.
y=x8y = x^{} -8

STEP 3

Next, we swap xx and yy. This means we replace every xx with yy and every yy with xx.
x=y38x = y^{3} -8

STEP 4

Now, we solve this equation for yy to get the inverse function. First, we add8 to both sides to isolate y3y^{3}.
x+8=y3x +8 = y^{3}

STEP 5

Finally, we take the cube root of both sides to solve for yy.
y=x+83y = \sqrt[3]{x +8}This is the inverse of the function f(x)f(x), which we denote as f1(x)f^{-1}(x).
f1(x)=x+83f^{-1}(x) = \sqrt[3]{x +8}The inverse of the function f(x)=x38f(x) = x^{3} -8 is f1(x)=x+83f^{-1}(x) = \sqrt[3]{x +8}.

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