Problem 36 - College Algebra

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 Given Information

To solve this problem, we need to understand the meaning of the function being one-to-one and the role of the inverse function. We are given that the function \(f\) is one-to-one, which means that every output value of the function corresponds to exactly one input value. Moreover, we know that \(f^{-1}(-2) = -1\), meaning that when the function \(f\) is inverted, an input of \(-2\) yields an output of \(-1\).

Step 2: Relate Inverse to Original Function

Since the function is one-to-one, we have a direct relationship between the inverse and the original function. Specifically, \(f^{-1}(-2) = -1\) tells us that \(f(-1) = -2\). This is because the inverse function \(f^{-1}\) mapping \(-2\) to \(-1\) implies the original function \(f\) maps \(-1\) to \(-2\).

Step 3: State the Solution

Now that we understand the relationship, the solution to the given problem is simply the evaluation of \(f(-1)\). We have determined \(f(-1) = -2\) based on the inverse relationship provided.

Key Concepts

One-to-One Function

A function is called a **one-to-one function** if every output value is linked to exactly one unique input value. This unique feature ensures that the function does not pair a single input with different outputs, making it injective.

• Imagine it as a scenario where every student has a unique locker, and no two students can share the same locker.
• To check if a function is one-to-one, we often use the horizontal line test on its graph. If any horizontal line crosses the function at more than one point, then the function is not one-to-one.

In terms of inverse functions, a one-to-one function is special because it guarantees the existence of an inverse function. This inverse effectively "undoes" the action of the original function. This means if you apply the original function to a one-to-one function's inverse, you get back your starting value.

Function Evaluation

**Function evaluation** involves finding the output of a function for a specific input. It is a fundamental concept that helps to understand how functions behave and map one value to another.

• When evaluating a function, substitute the given input value into the variable of the function's expression.
• Compute the expression to find the corresponding output.

For instance, to evaluate a one-to-one function at a particular point, you can substitute the known values into the inverse if provided, and use it to solve for unknowns. This process helps us understand function behavior and confirms the correspondence between inputs and outputs, which is crucial when determining inverse relationships.

Algebraic Functions

Algebraic functions are made up of algebraic expressions, which involve operations like addition, subtraction, multiplication, division, and taking roots. They are familiar from many basic algebraic manipulations and include polynomials, rational functions, and more.

• These functions can be defined explicitly with a formula or description.
• Understanding these functions is crucial for solving more complex mathematical problems and operations, such as finding inverses or evaluating them at specific points.

When dealing with one-to-one algebraic functions, finding inverses becomes simpler. By rearranging terms and using algebraic rules, you can derive the inverse function. This is possible if the function maintains a consistent input-output relationship, further emphasizing the significance of a one-to-one correspondence in inverse computations.