Math Problem Statement

The function

f left parenthesis x right parenthesis equals x cubed minus 2f(x)=x3−2

is​ one-to-one.

a. Find an equation for

f Superscript negative 1f−1​,

the inverse function.

b. Verify that your equation is correct by showing that

f left parenthesis f Superscript negative 1 Baseline left parenthesis x right parenthesis right parenthesisff−1(x)equals=x

and

f Superscript negative 1 Baseline left parenthesis f left parenthesis x right parenthesis right parenthesisf−1(f(x))equals=x.

Question content area bottom

Part 1

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 Superscript negative 1 Baseline left parenthesis x right parenthesisf−1(x)equals=RootIndex 3 StartRoot x plus 2 EndRoot3x+2​,

for all x

Your answer is correct.

B.

f Superscript negative 1 Baseline left parenthesis x right parenthesisf−1(x)equals=enter your response here​,

for

xgreater than or equals≥enter your response here

C.

f Superscript negative 1 Baseline left parenthesis x right parenthesisf−1(x)equals=enter your response here​,

for

xless than or equals≤enter your response here

D.

f Superscript negative 1 Baseline left parenthesis x right parenthesisf−1(x)equals=enter your response here​,

for

xnot equals≠enter your response here

Part 2

b. Verify that the equation is correct.

f left parenthesis f Superscript negative 1 Baseline left parenthesis x right parenthesis right parenthesisff−1(x)

equals=

fleft parenthesis nothing right parenthesisenter your response here

and

f Superscript negative 1 Baseline left parenthesis f left parenthesis x right parenthesis right parenthesisf−1(f(x))

equals=

f Superscript negative 1 Baseline left parenthesis nothing right parenthesisf−1enter your response here

Substitute.

equals=

enter your response here

equals=

enter your response here

Simplify.

Solution

Part 1: Finding the inverse function f−1(x)f^{-1}(x)

Given that the function f(x)=x3−2f(x) = x^3 - 2, we are tasked with finding its inverse.

To find the inverse of a function, we swap f(x)f(x) with yy and solve for xx:

y=x3−2y = x^3 - 2

  1. Add 2 to both sides to isolate x3x^3:

y+2=x3y + 2 = x^3

  1. Take the cube root of both sides to solve for xx:

x=y+23x = \sqrt[3]{y + 2}

Thus, the inverse function is:

f−1(x)=x+23f^{-1}(x) = \sqrt[3]{x + 2}

So, the correct answer is: A. f−1(x)=x+23\boxed{A. \ f^{-1}(x) = \sqrt[3]{x + 2}}

Part 2: Verifying the inverse function

We need to verify that:

  1. f(f−1(x))=xf(f^{-1}(x)) = x
  2. f−1(f(x))=xf^{-1}(f(x)) = x

Step 1: Verify f(f−1(x))=xf(f^{-1}(x)) = x

Substitute f−1(x)=x+23f^{-1}(x) = \sqrt[3]{x + 2} into the function f(x)=x3−2f(x) = x^3 - 2:

f(f−1(x))=f(x+23)f(f^{-1}(x)) = f\left(\sqrt[3]{x + 2}\right)

Now, apply the function f(x)=x3−2f(x) = x^3 - 2 to x+23\sqrt[3]{x + 2}:

[ f\left(\sqrt[3]{x + 2}\right) = \left(\sqrt[3]{x + 2}\right)^3 - 2 ]

Since the cube and cube root cancel out, we get:

f(f−1(x))=x+2−2=xf(f^{-1}(x)) = x + 2 - 2 = x

So, f(f−1(x))=xf(f^{-1}(x)) = x is verified.

Step 2: Verify f−1(f(x))=xf^{-1}(f(x)) = x

Substitute f(x)=x3−2f(x) = x^3 - 2 into f−1(x)=x+23f^{-1}(x) = \sqrt[3]{x + 2}:

f−1(f(x))=f−1(x3−2)f^{-1}(f(x)) = f^{-1}(x^3 - 2)

Now, apply the inverse function f−1(x)=x+23f^{-1}(x) = \sqrt[3]{x + 2}:

f−1(x3−2)=(x3−2)+23f^{-1}(x^3 - 2) = \sqrt[3]{(x^3 - 2) + 2}

Simplifying inside the cube root:

f−1(x3−2)=x33=xf^{-1}(x^3 - 2) = \sqrt[3]{x^3} = x

So, f−1(f(x))=xf^{-1}(f(x)) = x is also verified.

Conclusion:

Both verifications hold true, so the inverse function is correct.

Next Steps:

Would you like more details on this solution, or do you have any further questions?

Here are 5 related questions to expand on this:

  1. How do we determine whether a function is one-to-one?
  2. What are the general steps for finding the inverse of a cubic function?
  3. Can all cubic functions have an inverse? Why or why not?
  4. How does the domain and range of a function affect its inverse?
  5. How would you find the inverse if the original function had a different constant?

Tip: When checking the inverse of a function, always ensure that both f(f−1(x))=xf(f^{-1}(x)) = x and f−1(f(x))=xf^{-1}(f(x)) = x hold true to confirm correctness.

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Math Problem Analysis

Mathematical Concepts

Inverse Functions
Cubic Functions
Function Composition

Formulas

f(x) = x^3 - 2
f^(-1)(x) = (x + 2)^(1/3)

Theorems

Inverse Function Theorem
Function Composition Verification

Suitable Grade Level

Grades 11-12

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