Problem 8 - Intermediate Algebra

Question

If a function turns an input of 2 into an output of \(5,\) the inverse function will turn an input of 5 into the output

Step-by-Step Solution

Answer

The inverse function turns an input of 5 into the output 2.

Step 1: Understanding the Function Relationship

A function is a relationship where each input corresponds to exactly one output. Here, we know the function turns an input of 2 into an output of 5. Mathematically, it can be represented as \( f(2) = 5 \).

Step 2: Understanding the Inverse Function

The inverse of a function reverses the roles of inputs and outputs. For a function \( f \), the inverse \( f^{-1} \) satisfies the condition that \( f(f^{-1}(y)) = y \) and \( f^{-1}(f(x)) = x \).

Step 3: Applying the Inverse Function

Given \( f(2) = 5 \), to find the inverse function value \( f^{-1}(5) \), we use the fact that \( f(f^{-1}(5)) = 5 \). You need to determine what input to the function results in an output of 5, knowing it was originally the output from an input of 2.

Step 4: Concluding the Inverse Value

Since we established that \( f(2) = 5 \), to find \( f^{-1}(5) \), use the relationship \( f^{-1}(5) = 2 \) because 2 was the input that produced a 5.

Key Concepts

Function Relationship

When we talk about functions in mathematics, we are discussing a special type of relationship between two sets of things: inputs and outputs. Imagine this like a machine where you put something in, and something else comes out based on certain rules.

• In our example, a function takes an input of 2 and gives an output of 5.
• This is expressed mathematically as \( f(2) = 5 \).
• The key idea is that for each input, there is only one output. This one-to-one relationship helps us understand how these functions work.

Knowing this is crucial because it sets the stage for how we switch roles when dealing with inverse functions.

Mathematical Representation

Mathematical representation is about turning real-world problems into math equations. It allows us to use symbols to describe the relationships between numbers or objects, making it easier to manipulate or solve them. In our context, we see it in the form of function equations like \( f(x) \).

• When we say \( f(2) = 5 \), it means when the function \( f \) gets 2 as its input, it processes it to produce the output 5.
• This abstraction helps us handle complex relationships by simplifying them into understandable terms.

Understanding how to represent these relationships mathematically is the first step towards grasping more complex concepts like inverses.

Input-Output Roles

In every function, there are input and output roles that define what goes into the function and what comes out. Understanding these roles is essential to working with functions and their inverses.

• The input is the value you provide to the function - in our exercise, it was 2.
• The output is what you get after processing the input through the function - here, it was 5.
• The inverse function flips these roles, which means the input becomes the output, and the output becomes the input.

This flip-flop is pivotal to finding inverse functions, as it allows us to trace outputs back to their original inputs.

Understanding Inverse

An inverse function essentially reverses what a given function does. It takes an output from the original function and produces the input that originally generated that output.

• For the function \( f \), the inverse is denoted as \( f^{-1} \).
• If \( f(2) = 5 \), then \( f^{-1}(5) = 2 \), using the original input-output pair.
• This tells us that applying the inverse function to the output 5 gives us the input 2.

Understanding the concept of inverse functions helps in scenarios where you have an output and need to determine the corresponding input, thereby enhancing problem-solving capabilities.