Math Problem Statement

  1. π‘“βˆ’1(βˆ’4)=

  2. π‘“βˆ’1(βˆ’3)=

  3. π‘“βˆ’1(0)=

  4. π‘“βˆ’1(2)=

  5. π‘“βˆ’1(4)=

1. f^{-1}(-4)= 2. f^{-1}(-3)= 3. f^{-1}(0)= 4. f^{-1}(2)= 5. f^{-1}(4)=

Solution

Let's analyze the image, which appears to be a graph of a function f(x)f(x). To answer the questions regarding fβˆ’1(y)f^{-1}(y), or the inverse function, we need to find the xx-values for specific yy-values (since fβˆ’1(y)=xf^{-1}(y) = x if and only if f(x)=yf(x) = y).

Here’s the breakdown based on the graph:

  1. fβˆ’1(βˆ’4)f^{-1}(-4): This is the xx-value where y=βˆ’4y = -4. From the graph, it appears that the point where y=βˆ’4y = -4 corresponds to x=βˆ’1x = -1. So, fβˆ’1(βˆ’4)=βˆ’1f^{-1}(-4) = -1.

  2. fβˆ’1(βˆ’3)f^{-1}(-3): The xx-value where y=βˆ’3y = -3 corresponds to x=0x = 0 from the graph. So, fβˆ’1(βˆ’3)=0f^{-1}(-3) = 0.

  3. fβˆ’1(0)f^{-1}(0): The point where y=0y = 0 corresponds to x=2x = 2. Thus, fβˆ’1(0)=2f^{-1}(0) = 2.

  4. fβˆ’1(2)f^{-1}(2): For y=2y = 2, there doesn’t seem to be a corresponding point on the visible part of the graph, so it's likely undefined within this range.

  5. fβˆ’1(4)f^{-1}(4): For y=4y = 4, similar to y=2y = 2, there’s no corresponding xx-value on the visible graph.

To summarize:

  1. fβˆ’1(βˆ’4)=βˆ’1f^{-1}(-4) = -1
  2. fβˆ’1(βˆ’3)=0f^{-1}(-3) = 0
  3. fβˆ’1(0)=2f^{-1}(0) = 2
  4. fβˆ’1(2)f^{-1}(2) is undefined
  5. fβˆ’1(4)f^{-1}(4) is undefined

Would you like further details or clarification?

Here are five questions related to this graph and inverse functions:

  1. How is the domain of fβˆ’1(x)f^{-1}(x) determined from the original function?
  2. What does it mean for the function if fβˆ’1(x)f^{-1}(x) is undefined for certain values?
  3. How would the function’s behavior change if the graph extended beyond the current range?
  4. What properties must the function f(x)f(x) have to ensure that fβˆ’1(x)f^{-1}(x) exists?
  5. How would you calculate the inverse function fβˆ’1(x)f^{-1}(x) algebraically if given a formula for f(x)f(x)?

Tip: When looking at graphs of functions and their inverses, reflect the graph over the line y=xy = x to visualize the inverse relationship more easily!

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Math Problem Analysis

Mathematical Concepts

Inverse functions
Graph interpretation

Formulas

f^{-1}(y) = x if and only if f(x) = y

Theorems

Inverse function theorem

Suitable Grade Level

Grades 9-12