Problem 33 - College Algebra with Corequisite Support

Question

For the following exercises, evaluate or solve, assuming that the function \(f\) is one-to-one. If \(f(6)=7,\) find \(f^{-1}(7)\)

Step-by-Step Solution

Answer

\(f^{-1}(7) = 6\)

Step 1: Understand the Definition of Inverse Function

Since the function \(f\) is one-to-one and \(f(6) = 7\), the inverse function \(f^{-1}(x)\) will map \(7\) back to \(6\). This means that the inverse function \(f^{-1}\) reverses the effect of \(f\).

Step 2: Apply the Definition to Find the Inverse

To find \(f^{-1}(7)\), you need to identify the input value of the original function \(f(x)\) which gives the output \(7\). Based on the given information, \(f(6) = 7\), so the input value is \(6\).

Key Concepts

One-to-One Functions

In simple terms, a one-to-one function ensures that every unique input has a unique output. This property is essential for a function to have an inverse. Why is this important? Because if more than one input could lead to the same output, it would be impossible to determine the correct original input from a given output.

• Unique mapping: Every output is paired with exactly one input.
• Horizontal line test: If a horizontal line intersects the graph of the function at most once, the function is one-to-one.

Understanding the concept of a one-to-one function helps in grasping how inverses work, where the function 'undoes' itself.

Function Evaluation

Function evaluation involves plugging a value into the function and getting the result. For example, if you evaluate a function at 6, you apply the rules of the function to the number 6 to find out what output it produces.
In the exercise, we have that the evaluation of function \( f \) at 6 gives 7. In math terms, this is written as \( f(6) = 7 \). This is crucial information because it shows us precisely how the function behaves for a particular input, allowing us to later find the inverse or solve for unknowns.

Inverse Function Calculation

Calculating the inverse of a function involves finding a new function that reverses the actions of the original. When you apply the inverse function to the output of the original function, it should return the original input.
Here's a step-by-step:

• Acknowledge that a function \( f \) and its inverse \( f^{-1} \) satisfy the condition \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).
• Based on \( f(6) = 7 \), to find \( f^{-1}(7) \), consider the number that \( f \) was originally applied to in order to achieve 7. In this case, it is 6.

This shows the power and simplicity of inverses in solving for unknown inputs when the output is known.

Understanding Inverses

Understanding inverses requires imagining a function working in reverse. If a function turns sets of inputs into specific outputs, its inverse does the opposite.
Why are inverses useful?

• They allow prediction of inputs based on outputs.
• They enable solving equations that involve the original function.

For instance, knowing that \( f(6) = 7 \) lets us determine that \( f^{-1}(7) = 6 \). This is because the inverse function precisely counteracts the original function's effect.