Math  /  Algebra

QuestionFind the inverse function of f(x)=4+x3f(x)=4+\sqrt[3]{x} f1(x)=f^{-1}(x)= \square

Studdy Solution

STEP 1

1. The function f(x)=4+x3f(x) = 4 + \sqrt[3]{x} is bijective (one-to-one and onto), allowing us to find its inverse.
2. To find the inverse function f1(x)f^{-1}(x), we need to solve the equation y=4+x3y = 4 + \sqrt[3]{x} for xx in terms of yy.
3. The inverse function f1(x)f^{-1}(x) will reverse the operations performed by f(x)f(x).

STEP 2

1. Replace f(x)f(x) with yy and solve for xx in terms of yy.
2. Express the solution x=g(y)x = g(y), where g(y)g(y) is the inverse function.
3. Replace yy with xx to write the final form of f1(x)f^{-1}(x).

STEP 3

Start with the equation y=4+x3y = 4 + \sqrt[3]{x}.
y=4+x3 y = 4 + \sqrt[3]{x}

STEP 4

Subtract 4 from both sides to isolate the cube root term.
y4=x3 y - 4 = \sqrt[3]{x}

STEP 5

Cube both sides to solve for xx in terms of yy.
(y4)3=x (y - 4)^3 = x

STEP 6

Replace yy with xx to write the inverse function f1(x)f^{-1}(x).
f1(x)=(x4)3 f^{-1}(x) = (x - 4)^3
Thus, the inverse function of f(x)=4+x3f(x) = 4 + \sqrt[3]{x} is: f1(x)=(x4)3f^{-1}(x) = (x-4)^3

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