Math Problem Statement

Given the function f(x) = ∛x + 12, (a) Find f^(-1)(x), (b) Graph f and f^(-1) in the same rectangular coordinate system.

Solution

Let's work through the given problem step by step.

Given:

The function is f(x)=x3+12f(x) = \sqrt[3]{x} + 12.

Part (a) Find f1(x)f^{-1}(x), the inverse of the function.

To find the inverse of f(x)f(x), follow these steps:

  1. Start by setting f(x)=yf(x) = y: y=x3+12y = \sqrt[3]{x} + 12
  2. Solve for xx in terms of yy:
    • First, subtract 12 from both sides: y12=x3y - 12 = \sqrt[3]{x}
    • Cube both sides to remove the cube root: (y12)3=x(y - 12)^3 = x
  3. Now, express xx as a function of yy: f1(x)=(x12)3f^{-1}(x) = (x - 12)^3

Thus, the inverse function is: f1(x)=(x12)3f^{-1}(x) = (x - 12)^3

Part (b) Choose the correct graph.

You need to select the graph that shows both the original function f(x)=x3+12f(x) = \sqrt[3]{x} + 12 and its inverse f1(x)=(x12)3f^{-1}(x) = (x - 12)^3. In general:

  • The graph of a function and its inverse will be symmetric about the line y=xy = x.
  • The original function f(x)=x3+12f(x) = \sqrt[3]{x} + 12 is a shifted cube root graph, and its inverse f1(x)=(x12)3f^{-1}(x) = (x - 12)^3 is a shifted cubic graph.

From the images:

  • The original function (green curve) should resemble a cube root curve shifted up by 12 units.
  • The inverse function (red curve) should look like a cubic function shifted to the right by 12 units.

Based on the symmetry and the behavior of the functions, the correct graph appears to be Option D.

Would you like further details or clarification?

Follow-up Questions:

  1. Why does the cube root function have a different shape than a cubic function?
  2. How do you confirm if a graph represents an inverse function?
  3. How do transformations like shifting affect the graph of a function?
  4. Why is the line y=xy = x significant for graphs of inverses?
  5. How would you graph f1(x)f^{-1}(x) if given different functions?

Tip:

When finding an inverse function, always swap xx and yy and solve for yy, which will be your new function f1(x)f^{-1}(x).

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Math Problem Analysis

Mathematical Concepts

Functions
Inverses of Functions
Cubic and Cube Root Functions

Formulas

f(x) = ∛x + 12
Inverse of a function: swap x and y, then solve for y
f^(-1)(x) = (x - 12)^3

Theorems

Symmetry of a function and its inverse across the line y = x

Suitable Grade Level

Grades 9-12