Math Problem Statement

how to solve d

How to solve (g(2))^-1 using the given table of functions f and g.

Solution

To solve part (d), (g(2))1(g(2))^{-1}, we need to determine the value of g(2)g(2) from the table, then find the inverse value corresponding to that result for g1g^{-1}.

Step-by-Step Solution:

  1. Find g(2)g(2):
    Look up the value of g(x)g(x) when x=2x = 2 in the table. From the table,
    g(2)=3.g(2) = 3.

  2. Find the inverse g1(3)g^{-1}(3):
    Now, we need to find the value of xx such that g(x)=3g(x) = 3. From the table,
    g(x)=3 when x=2.g(x) = 3 \text{ when } x = 2.

    Therefore,
    g1(3)=2.g^{-1}(3) = 2.

Conclusion:

The value of (g(2))1(g(2))^{-1} is 22.

If you have any further questions or want more details, feel free to ask!

Related Questions:

  1. What is the general method for finding inverses of functions?
  2. How do you compute compositions of functions using tables?
  3. What are some examples of functions where the inverse is not well-defined?
  4. Can you explain how to determine whether a function is invertible from a table?
  5. What are some practical applications of inverse functions in real life?

Tip:

Always check the domain and range of a function to ensure the inverse is valid and within the correct values.

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Math Problem Analysis

Mathematical Concepts

Functions
Function Inverses
Composition of Functions

Formulas

g(x) = y, g^{-1}(y) = x

Theorems

Inverse function theorem
Properties of invertible functions

Suitable Grade Level

Grades 9-12