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: $g^{-1}(-1)=9$ $g^{-1}(-2)=1$ $g^{-1}(9)=4$ $g^{-1}(-1)=-2$

Solution

Solution Steps

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

Step 1: Understanding the Inverse Function

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

Step 2: Evaluating Each Statement

Using the known values:

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

Now we can evaluate the statements:

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