Math Problem Statement

The function f(x) = x^3 - 5 is one-to-one. (a) Find the inverse of f and check the answer. (b) Find the domain and the range of f and f^{-1}. (c) Graph f, f^{-1}, and y = x on the same coordinate axes.

Solution

Let's work through this question step by step:

The function f(x)=x35f(x) = x^3 - 5 is one-to-one.

(a) Finding the inverse of f(x)f(x)

  1. Start with y=x35y = x^3 - 5.
  2. Solve for xx in terms of yy: y+5=x3    x=y+53.y + 5 = x^3 \quad \implies \quad x = \sqrt[3]{y + 5}.
  3. Swap xx and yy to find the inverse function: f1(x)=x+53.f^{-1}(x) = \sqrt[3]{x + 5}.

So, the inverse function is f1(x)=x+53f^{-1}(x) = \sqrt[3]{x + 5}.

(b) Domain and range of f(x)f(x) and f1(x)f^{-1}(x)

  • Domain of f(x)f(x): Since x35x^3 - 5 is a cubic function, it is defined for all real numbers xx. Therefore: Domain of f(x)=(,).\text{Domain of } f(x) = (-\infty, \infty).

  • Range of f(x)f(x): A cubic function is surjective (covers all real numbers). Therefore: Range of f(x)=(,).\text{Range of } f(x) = (-\infty, \infty).

  • Domain of f1(x)f^{-1}(x): The domain of f1(x)f^{-1}(x) is the range of f(x)f(x), which is: Domain of f1(x)=(,).\text{Domain of } f^{-1}(x) = (-\infty, \infty).

  • Range of f1(x)f^{-1}(x): The range of f1(x)f^{-1}(x) is the domain of f(x)f(x), which is: Range of f1(x)=(,).\text{Range of } f^{-1}(x) = (-\infty, \infty).

(c) Graph f(x)f(x), f1(x)f^{-1}(x), and y=xy = x on the same coordinate axes

To graph these:

  1. f(x)=x35f(x) = x^3 - 5 is a cubic function shifted down by 5.
  2. f1(x)=x+53f^{-1}(x) = \sqrt[3]{x + 5} is the reflection of f(x)f(x) over y=xy = x.
  3. y=xy = x is the line of symmetry.

Would you like me to sketch this graph for you?


Questions to Expand Learning:

  1. What are the key properties of one-to-one functions, and why are they necessary for finding an inverse?
  2. How would the inverse function change if f(x)=x3+cf(x) = x^3 + c instead of x35x^3 - 5?
  3. How can we verify if two functions are inverses of each other?
  4. What happens to the domain and range when you apply transformations to a cubic function, such as shifting or reflecting?
  5. Why is y=xy = x the line of symmetry for inverse functions?

Tip:

When solving for an inverse function, always verify your solution by checking if f(f1(x))=xf(f^{-1}(x)) = x and f1(f(x))=xf^{-1}(f(x)) = x.

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Math Problem Analysis

Mathematical Concepts

Inverse Functions
Domain and Range
Graphing Functions
One-to-One Functions

Formulas

Inverse function: y = f^{-1}(x)
Cubic root: x = \sqrt[3]{y}

Theorems

A one-to-one function has an inverse
Inverse functions reflect over y = x

Suitable Grade Level

Grades 10-12