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Math  /  Algebra

QuestionThe function f(x)=x3−3f(x)=x^{3}-3 is one-to-one. a. Find an equation for f−1f^{-1}, 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 all xx C. f−1(x)=f^{-1}(x)= □\square , for x≤x \leq □\square D. f−1(x)=f^{-1}(x)= □\square , for x≠x \neq □\square

Studdy Solution

STEP 1

1. The function f(x)=x3−3 f(x) = x^3 - 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 should 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. Solve for the inverse function f−1(x) f^{-1}(x) .
2. Verify the inverse function by checking f(f−1(x))=x f(f^{-1}(x)) = x .
3. Verify the inverse function by checking f−1(f(x))=x f^{-1}(f(x)) = x .
4. Determine the correct choice for the inverse function.

STEP 3

To find the inverse function, start by setting y=f(x)=x3−3 y = f(x) = x^3 - 3 . We need to solve for x x in terms of y y .
y=x3−3 y = x^3 - 3
Add 3 to both sides:
y+3=x3 y + 3 = x^3
Take the cube root of both sides to solve for x x :
x=y+33 x = \sqrt[3]{y + 3}
Thus, the inverse function is:
f−1(x)=x+33 f^{-1}(x) = \sqrt[3]{x + 3}

STEP 4

Verify that 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(x+33)=(x+33)3−3 f(f^{-1}(x)) = f(\sqrt[3]{x + 3}) = (\sqrt[3]{x + 3})^3 - 3
Simplify:
=(x+3)−3=x = (x + 3) - 3 = x
This confirms that f(f−1(x))=x f(f^{-1}(x)) = x .

STEP 5

Verify that 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(x3−3)=(x3−3)+33 f^{-1}(f(x)) = f^{-1}(x^3 - 3) = \sqrt[3]{(x^3 - 3) + 3}
Simplify:
=x33=x = \sqrt[3]{x^3} = x
This confirms that f−1(f(x))=x f^{-1}(f(x)) = x .

STEP 6

Determine the correct choice for the inverse function.
Since the function f(x)=x3−3 f(x) = x^3 - 3 is a cubic function and is defined for all real numbers, the inverse function f−1(x)=x+33 f^{-1}(x) = \sqrt[3]{x + 3} is also defined for all real numbers.
Thus, the correct choice is:
B. f−1(x)=x+33 f^{-1}(x) = \sqrt[3]{x + 3} , for all x x
The inverse function is x+33 \boxed{\sqrt[3]{x + 3}} for all x x .

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