Problem 36 - College Algebra with Corequisite Support

Question

For the following exercises, evaluate or solve, assuming that the function \(f\) is one-to-one. If \(f^{-1}(-2)=-1,\) find \(f(-1)\)

Step-by-Step Solution

Answer

\(f(-1) = -2\)

Step 1: Understanding the given information

We are given that the inverse function, \(f^{-1}(-2) = -1\). This means that when the function \(f\) is evaluated at \(-1\), it gives the output \(-2\). This relationship is because \(f^{-1}\) reverses the operation of \(f\).

Step 2: Determine the function value

Based on the fact that \(f^{-1}(-2) = -1\) implies \(f(-1) = -2\), we can directly conclude \(f(-1) = -2\). The inverse relationship is confirmed by the one-to-one property of the function.

Key Concepts

One-to-one function

In mathematics, an important concept to understand is the idea of a one-to-one function. This type of function has a unique property: every output value is paired with exactly one input value. Consequently, no two different input values will produce the same output.
This concept is incredibly crucial when dealing with inverse functions, as only one-to-one functions can have inverses. To recognize a function as one-to-one, a common method is the "Horizontal Line Test." If no horizontal line intersects the graph of a function at more than one point, then the function is one-to-one.
If you've figured out that

• Each input has a distinct output,
• No repeated y-values for different x-values,

then you are dealing with a one-to-one function, making it a candidate for having an inverse.

Function evaluation

Function evaluation is the process of calculating the output of a function for a specific input. It is similar to the concept of plugging numbers into a formula. Suppose you're given a function \( f(x) \). When you evaluate \( f(x) \) for an input \( x = a \), you are basically finding out what \( f(a) \) is.
The goal is to examine how certain functions transform inputs to outputs. In our original problem, though the function \( f \) is not explicitly defined, you can determine that for the input \( -1 \), the output is \( -2 \).
Knowing how to evaluate functions efficiently allows us to work back and forth between a function and its inverse smoothly, which makes function evaluation a key part of understanding and using inverse functions effectively. Here, understanding that \( f(-1) = -2 \) ensures the given inverse information can be verified.

Inverse relationship

The inverse relationship between two functions refers to how they essentially reverse each other’s operations. When we have a function \( f(x) \) and its inverse \( f^{-1}(y) \), they satisfy the condition that applying one after the other results in the original input. In simpler terms, \( f(f^{-1}(y)) = y \) and \( f^{-1}(f(x)) = x \).
In the original exercise, you're provided the information \( f^{-1}(-2) = -1 \). This means that when you apply function \( f \) to \(-1\), you get \(-2\). So essentially, these functions undo each other's actions.
Understanding this inverse relationship is essential in puzzles where you need to find either the original function value or its inverse based on limited clues. Without this, it would be impossible to leverage one piece of information to extract the other, highlighting the significance of the inverse relationship in mathematical operations.