Math Problem Statement

Find the inverse functions of the following one-to-one functions: Example:

Step l = y = x + 3

Step 2. x = y + 3

Step 3 = x - 3 = y or y = x - 3

Step 4 = f - 1(x) = x - 3

Step 1. Replace f(x) by y)

Step 2. (Interchange x's and y's)

Step 3. (Solve for y in step 2)

Step 4. (Replace y with f-1)

Solve this 3:

  1. f(x) = (x + 3)^3
  2. f(x) = 1/2 x + 4
  3. f(x) = 4

Solution

Let's follow the steps for finding the inverse of each function you provided.

1. f(x)=(x+3)3f(x) = (x + 3)^3

Step 1: Replace f(x)f(x) with yy: y=(x+3)3y = (x + 3)^3

Step 2: Interchange xx and yy: x=(y+3)3x = (y + 3)^3

Step 3: Solve for yy: x3=y+3\sqrt[3]{x} = y + 3 y=x33y = \sqrt[3]{x} - 3

Step 4: Replace yy with f1(x)f^{-1}(x): f1(x)=x33f^{-1}(x) = \sqrt[3]{x} - 3


2. f(x)=12x+4f(x) = \frac{1}{2}x + 4

Step 1: Replace f(x)f(x) with yy: [ y = \frac{

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Inverse Functions
Algebra
Cubic Functions
Linear Equations

Formulas

Inverse of f(x): If y = f(x), solve for x to find f^{-1}(x)
For linear functions: y = mx + b
For cubic functions: f(x) = (x + a)^n

Theorems

Inverse Function Theorem
Properties of Linear Functions
Properties of Cubic Functions

Suitable Grade Level

Grades 9-12