Math Problem Statement

State the inverse function g(x) of this equation: g(x)=3x-2

Solution

To find the inverse of the function g(x)=3x2g(x) = 3x - 2, we follow these steps:

  1. Rewrite the function using yy instead of g(x)g(x): y=3x2y = 3x - 2

  2. Swap xx and yy to begin the process of finding the inverse: x=3y2x = 3y - 2

  3. Solve for yy (which will be g1(x)g^{-1}(x)): x+2=3yx + 2 = 3y y=x+23y = \frac{x + 2}{3}

Thus, the inverse function is: g1(x)=x+23g^{-1}(x) = \frac{x + 2}{3}

Would you like any further details or explanations?

Here are five related questions to further explore the concept of inverse functions:

  1. How can you verify if two functions are inverses of each other?
  2. What happens when you compose a function with its inverse?
  3. How would the graph of g1(x)g^{-1}(x) compare to the graph of g(x)g(x)?
  4. Can every function have an inverse? Why or why not?
  5. How does the domain and range of a function affect its inverse?

Tip: To check if two functions are inverses, you can compose them and see if the result is the identity function: f(g(x))=xf(g(x)) = x and g(f(x))=xg(f(x)) = x.

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Math Problem Analysis

Mathematical Concepts

Algebra
Functions
Inverse Functions

Formulas

Inverse function formula: g^{-1}(x)
g(x) = 3x - 2
g^{-1}(x) = (x + 2) / 3

Theorems

Inverse Function Theorem

Suitable Grade Level

Grades 9-11