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Math  /  Algebra

QuestionThe function f(x)=(x+7)3f(x)=(x+7)^{3} is one-to-one. a. Find an equation for f−1(x)f^{-1}(x), the inverse function. b. Verify that your equation is correct by showing that f(f−1(x))=xf\left(f^{-1}(x)\right)=x and f−1(f(x))=xf^{-1}(f(x))=x. a. Select the correct choice below and fill in the answer box(es) to complete your choice. (Simplify your answer. Use integers or fractions for any numbers in the expression.) A. f−1(x)=f^{-1}(x)= □\square , for x≥x \geq □\square B. f−1(x)=f^{-1}(x)= □\square , for x≤\mathrm{x} \leq □\square C. f−1(x)=x13−7f^{-1}(x)=x^{\frac{1}{3}}-7, for all xx D. f−1(x)=f^{-1}(x)= □\square , for x≠x \neq □\square b. Verify that the equation is correct. f(f−1(x))=f(x3−7) and f−1(f(x))=f−1((x+7)3) Substitute. =x=−1x Simplify \begin{array}{rlrlrl} f\left(f^{-1}(x)\right) & =f(\sqrt[3]{x}-7) & \text { and } & f^{-1}(f(x)) & =f^{-1}\left((x+7)^{3}\right) & \text { Substitute. } \\ & =x & & ={ }^{-1} x & \text { Simplify } \end{array}
The equation is □\square

Studdy Solution

STEP 1

1. The function f(x)=(x+7)3 f(x) = (x+7)^3 is one-to-one, meaning it has an inverse.
2. To find the inverse, we need to solve for x x in terms of y y where y=f(x) y = f(x) .
3. The inverse function, f−1(x) f^{-1}(x) , will satisfy f(f−1(x))=x f(f^{-1}(x)) = x and f−1(f(x))=x f^{-1}(f(x)) = x .

STEP 2

1. Find the inverse function f−1(x) f^{-1}(x) .
2. Verify the inverse by checking f(f−1(x))=x f(f^{-1}(x)) = x .
3. Verify the inverse by checking f−1(f(x))=x f^{-1}(f(x)) = x .

STEP 3

To find the inverse function f−1(x) f^{-1}(x) , start by setting y=f(x)=(x+7)3 y = f(x) = (x+7)^3 .
y=(x+7)3 y = (x+7)^3

STEP 4

Solve for x x in terms of y y :
y=(x+7)3 y = (x+7)^3
Take the cube root of both sides:
y3=x+7 \sqrt[3]{y} = x + 7

STEP 5

Isolate x x :
x=y3−7 x = \sqrt[3]{y} - 7
Thus, the inverse function is:
f−1(x)=x3−7 f^{-1}(x) = \sqrt[3]{x} - 7

STEP 6

Verify the inverse by checking f(f−1(x))=x f(f^{-1}(x)) = x :
Substitute f−1(x) f^{-1}(x) into f(x) f(x) :
f(f−1(x))=f(x3−7)=((x3−7)+7)3 f(f^{-1}(x)) = f(\sqrt[3]{x} - 7) = ((\sqrt[3]{x} - 7) + 7)^3
Simplify:
=(x3)3=x = (\sqrt[3]{x})^3 = x

STEP 7

Verify the inverse by checking f−1(f(x))=x f^{-1}(f(x)) = x :
Substitute f(x) f(x) into f−1(x) f^{-1}(x) :
f−1(f(x))=f−1((x+7)3)=(x+7)33−7 f^{-1}(f(x)) = f^{-1}((x+7)^3) = \sqrt[3]{(x+7)^3} - 7
Simplify:
=x+7−7=x = x + 7 - 7 = x
The correct choice is C: f−1(x)=x13−7 f^{-1}(x) = x^{\frac{1}{3}} - 7 , for all x x .
The equation is verified as correct.

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