Math Problem Statement

Use the given graph of y = f(x) to evaluate the following: (a) f(-3), (b) f(2), (c) f^(-1)(1), (d) f^(-1)(-3).

Solution

The image provided shows a question based on the graph of a function f(x)f(x). To evaluate the answers, we can read the function values from the graph.

The points provided on the graph are:

  • f(3)=9f(-3) = -9
  • f(2)=5f(2) = 5
  • f(1)=2f(1) = 2
  • f1(1)=2f^{-1}(1) = -2 (this is the x-value where f(x)=1f(x) = 1)
  • f1(3)=3f^{-1}(-3) = 3 (this is the x-value where f(x)=3f(x) = -3)

Let's break it down:

  1. (a) f(3)f(-3): From the graph, when x=3x = -3, f(3)=9f(-3) = -9.
  2. (b) f(2)f(2): From the graph, when x=2x = 2, f(2)=5f(2) = 5.
  3. (c) f1(1)f^{-1}(1): The inverse asks for the x-value when f(x)=1f(x) = 1, which is x=2x = -2.
  4. (d) f1(3)f^{-1}(-3): The inverse asks for the x-value when f(x)=3f(x) = -3, which is x=3x = 3.

Now, you can proceed to fill in the answers for each part of the question:

  • (a) f(3)=9f(-3) = -9
  • (b) f(2)=5f(2) = 5
  • (c) f1(1)=2f^{-1}(1) = -2
  • (d) f1(3)=3f^{-1}(-3) = 3

Would you like more details on inverse functions or any other concept?

Here are some related questions:

  1. What is the graphical interpretation of an inverse function?
  2. How can you verify if a function is one-to-one and thus invertible?
  3. Can a function have an inverse if it is not one-to-one?
  4. How does the horizontal line test help in determining invertibility?
  5. What are some examples of real-world applications of inverse functions?

Tip: Inverse functions swap the input and output values of the original function, so visually, they reflect across the line y=xy = x.

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Math Problem Analysis

Mathematical Concepts

Functions
Inverse Functions
Graph Interpretation

Formulas

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Theorems

Inverse Function Theorem

Suitable Grade Level

Grades 9-12