Question

the xs and ys are the same
the xs and ys are made negative
the xs and ys are switched
the xs and ys are divided by 2
what does it mean to find the inverse of a function?

Explanation:

Step1: Recall the concept of inverse function

The inverse of a function \(y = f(x)\) is found by swapping the \(x\) and \(y\) variables. For example, if we have a function \(y = 2x+ 3\), to find its inverse, we first write \(x = 2y + 3\) (swap \(x\) and \(y\)) and then solve for \(y\).

Step2: Analyze each option

• Option 1: "the \(x\)'s and \(y\)'s are the same" is incorrect. If \(x\) and \(y\) are the same, it would imply \(y=x\) for all points, which is only the case for the function \(y = x\) itself, not the general process of finding an inverse.
• Option 2: "the \(x\)'s and \(y\)'s are made negative" is incorrect. Making \(x\) and \(y\) negative (\(y=-x\)) is a reflection over the \(y = -x\) line and not related to the inverse - function operation.
• Option 3: "the \(x\)'s and \(y\)'s are switched" is correct. By the definition of an inverse function, if \(F:X

ightarrow Y\) with \(y = F(x)\), then the inverse function \(F^{-1}:Y
ightarrow X\) is such that if we start with \(y = F(x)\), we rewrite it as \(x = F(y)\) (swapping \(x\) and \(y\)) and then solve for \(y\) to get \(y = F^{-1}(x)\).

• Option 4: "the \(x\)'s and \(y\)'s are divided by 2" is incorrect. Dividing \(x\) and \(y\) by 2 (\(y=\frac{x}{2}\)) is a linear transformation (a compression) and not the process of finding an inverse.

Answer:

the \(x\)'s and \(y\)'s are switched