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
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.
Final Answer
The only true statement is g−1(−1)=−2. Therefore, the answer is: