Math Problem Statement

The function

f left parenthesis x right parenthesis equals x cubed minus 10f(x)=x3−10

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=enter your response here​,

for

xnot equals≠enter your response here

B.

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

C.

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

D.

f Superscript negative 1 Baseline left parenthesis x right parenthesisf−1(x)equals=RootIndex 3 StartRoot x plus 10 EndRoot3x+10​,

for all x

Your answer is correct.

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)

We are given the function:

f(x)=x3−10f(x) = x^3 - 10

To find the inverse, follow these steps:

  1. Rewrite the function by replacing f(x)f(x) with yy: y=x3−10y = x^3 - 10

  2. Solve for xx in terms of yy: y+10=x3y + 10 = x^3 x=y+103x = \sqrt[3]{y + 10}

  3. Replace yy with xx to express the inverse function: f−1(x)=x+103f^{-1}(x) = \sqrt[3]{x + 10}

Thus, the inverse function is:

f−1(x)=x+103f^{-1}(x) = \sqrt[3]{x + 10}

Part 2: Verifying the Inverse

We need to verify that the inverse satisfies the following two properties:

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

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

Start with f−1(x)=x+103f^{-1}(x) = \sqrt[3]{x + 10}. Now substitute this into the function f(x)=x3−10f(x) = x^3 - 10:

f(f−1(x))=f(x+103)f(f^{-1}(x)) = f\left( \sqrt[3]{x + 10} \right)

Now apply the definition of ff:

[ f\left( \sqrt[3]{x + 10} \right) = \left( \sqrt[3]{x + 10} \right)^3 - 10 = (x + 10) - 10 = x ]

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

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

Now, start with f(x)=x3−10f(x) = x^3 - 10 and substitute it into the inverse function f−1(x)=x+103f^{-1}(x) = \sqrt[3]{x + 10}:

f−1(f(x))=f−1(x3−10)f^{-1}(f(x)) = f^{-1}(x^3 - 10)

Substitute x3−10x^3 - 10 into the inverse function:

[ f^{-1}(x^3 - 10) = \sqrt[3]{(x^3 - 10) + 10} = \sqrt[3]{x^3} = x ]

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

Final Answer:

  • Part (a): The inverse function is: f−1(x)=x+103f^{-1}(x) = \sqrt[3]{x + 10} This is valid for all values of xx.

  • Part (b): The two properties are verified: f(f−1(x))=xf(f^{-1}(x)) = x f−1(f(x))=xf^{-1}(f(x)) = x

Additional Exploration:

  1. How do we find the inverse of functions involving other operations like exe^x, ln⁡(x)\ln(x), or trigonometric functions?
  2. What happens if we have a function that is not one-to-one? Can it have an inverse?
  3. How do the graphs of a function and its inverse relate geometrically?
  4. What is the relationship between the domains and ranges of a function and its inverse?
  5. How can you check if a function is one-to-one?

Tip: When finding the inverse of a function, always make sure the original function is one-to-one, meaning each xx-value corresponds to exactly one yy-value, ensuring the inverse exists.

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Math Problem Analysis

Mathematical Concepts

Inverse Functions
Algebra
One-to-One Functions
Cubic Functions

Formulas

f^{-1}(x) = ∛(x + 10)
f(x) = x^3 - 10

Theorems

Inverse Function Theorem
One-to-One Function Property

Suitable Grade Level

Grades 9-11

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