Problem 35 - College Algebra with Corequisite Support

Question

For the following exercises, evaluate or solve, assuming that the function \(f\) is one-to-one. If \(f^{-1}(-4)=-8\), find \(f(-8)\).

Step-by-Step Solution

Answer

\( f(-8) = -4 \).

Step 1: Understand the Inverse Function Property

The function given is one-to-one, which means that it has an inverse. The property of inverse functions is such that if \( f^{-1}(b) = a \), then \( f(a) = b \). This means that performing both the function and its inverse returns the original input/output pair.

Step 2: Apply the Inverse Property to Find the Answer

Given \( f^{-1}(-4) = -8 \), we can use the property of inverse functions. This implies \( f(-8) = -4 \), because the inverse undoes the function's action.

Key Concepts

One-to-One Function

When dealing with inverse functions, understanding the concept of a one-to-one function is essential. A one-to-one function means that each input value maps to exactly one unique output value and vice versa. This characteristic ensures that a function has an inverse that is also a function.

A function is tested for one-to-one compatibility through transformations like the horizontal line test. Here’s how it works:

• Draw horizontal lines across the graph of the function.
• If any horizontal line crosses the graph more than once, the function is not one-to-one.

Such functions are pivotal because they invert perfectly, maintaining consistent input-output pairs and allowing us to apply inverse properties smoothly.

Function Evaluation

Evaluating a function means finding the output for a given input. It’s a simple process usually involving substitution of a value into the function's equation. For example, if you have a function like \( f(x) = 2x + 3 \), and you need to find \( f(2) \):

• Substitute 2 into the equation: \( f(2) = 2(2) + 3 \).
• This gives \( f(2) = 4 + 3 = 7 \).

Through function evaluation, you determine specific output values, aiding in understanding how the function behaves for different inputs. This step is crucial when validating facts about functions or verifying conditions for inverse evaluations.

Inverse Function Property

The inverse function property is a fundamental aspect when working with inverse functions. This property ensures that if a function \( f \) has an inverse \( f^{-1} \), they satisfy the condition:

• \( f(f^{-1}(x)) = x \) for all \( x \) in the domain of \( f^{-1} \)
• \( f^{-1}(f(x)) = x \) for all \( x \) in the domain of \( f \)

In practical terms, this means that applying a function and its inverse should bring you back to your original input.

In the original exercise, where given \( f^{-1}(-4) = -8 \), we accurately apply this property to determine that the inverse takes \(-8\) back to \(-4\), hence concluding that \( f(-8) = -4 \). This inherent two-step check ensures consistent and verifiable outcomes when dealing with inverse functions.