Questions: Consider the function g, which is a one-to-one function with values g(9)=-4 and g(-2)=-1. Which of the following must be true? Select all correct answers. Select all that apply: g^(-1)(-1)=9 g^(-1)(-2)=1 g^(-1)(9)=4 g^(-1)(-1)=-2

Consider the function g, which is a one-to-one function with values g(9)=-4 and g(-2)=-1.
Which of the following must be true?
Select all correct answers.

Select all that apply:
g^(-1)(-1)=9
g^(-1)(-2)=1
g^(-1)(9)=4
g^(-1)(-1)=-2
Transcript text: Consider the function $g$, which is a one-to-one function with values $g(9)=-4$ and $g(-2)=-1$. Which of the following must be true? Select all correct answers. Select all that apply: $g^{-1}(-1)=9$ $g^{-1}(-2)=1$ $g^{-1}(9)=4$ $g^{-1}(-1)=-2$
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Solution

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Solution Steps

To determine which statements are true, we need to understand the properties of the inverse function g1 g^{-1} . Specifically, if g(a)=b g(a) = b , then g1(b)=a g^{-1}(b) = a . Using this property, we can evaluate each statement.

Step 1: Understanding the Inverse Function

Given the function g g with values g(9)=4 g(9) = -4 and g(2)=1 g(-2) = -1 , we can determine the corresponding values of the inverse function g1 g^{-1} . The property of inverse functions states that if g(a)=b g(a) = b , then g1(b)=a g^{-1}(b) = a .

Step 2: Evaluating Each Statement

Using the known values:

  • From g(9)=4 g(9) = -4 , we have g1(4)=9 g^{-1}(-4) = 9 .
  • From g(2)=1 g(-2) = -1 , we have g1(1)=2 g^{-1}(-1) = -2 .

Now we can evaluate the statements:

  1. g1(1)=9 g^{-1}(-1) = 9 is False because g1(1)=2 g^{-1}(-1) = -2 .
  2. g1(2)=1 g^{-1}(-2) = 1 is False because there is no value a a such that g(a)=2 g(a) = -2 .
  3. g1(9)=4 g^{-1}(9) = 4 is False because there is no value b b such that g(b)=9 g(b) = 9 .
  4. g1(1)=2 g^{-1}(-1) = -2 is True.

Final Answer

The only true statement is g1(1)=2 g^{-1}(-1) = -2 . Therefore, the answer is:

g1(1)=2 \boxed{g^{-1}(-1) = -2}

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