Problem 34 - A Graphical Approach to College Algebra

Question

If a function \(f\) has an inverse and \(f(-3)=6,\) then \(f^{-1}(6)=\)____.

Step-by-Step Solution

Answer

\( f^{-1}(6) = -3 \).

Step 1: Understanding the Given Information

We know that a function \( f \) has an inverse, meaning it is a one-to-one function. We are given \( f(-3) = 6 \).

Step 2: Definition of Inverse Function

By definition, if \( f(a) = b \), then the inverse function \( f^{-1}(b) = a \). This relationship will help us find \( f^{-1}(6) \).

Step 3: Applying the Definition

Using the definition of inverse, since \( f(-3) = 6 \), it follows that \( f^{-1}(6) = -3 \).

Step 4: Final Result

Thus, the value of \( f^{-1}(6) \) is \(-3\).

Key Concepts

One-to-One Functions

One-to-one functions are an essential concept in understanding inverse functions. A function is considered one-to-one if every output is associated with exactly one unique input. This unique pairing ensures that the function passes the horizontal line test. Think of a function as a kind of machine that pairs every input exactly once with an output. If two different inputs can produce the same output, the function is not one-to-one and thus cannot have an inverse.

For a function to be one-to-one and thus invertible, it helps to visualize:

• Picture every possible output along the vertical axis.
• For every point you draw horizontally, there should be only one line intersecting your curve.

This is crucial because only functions that maintain this unique pairing for all inputs and outputs can be reversed or inverted. That's why one-to-one functions are so closely related to inverse functions.

Function Definition

At its core, a function is a rule that assigns outputs to inputs. It's like a recipe that produces the same result every time you use the same ingredients. In mathematical terms, a function maps each input to exactly one output, which can be represented as a set of ordered pairs \((x,y)\). Here, every 'x' value (input) is paired with one 'y' value (output).

To break it down further:

• Inputs are represented by the domain: all possible values of 'x'.
• Outputs form the range: all possible values of 'y'.

The beauty of functions lies in their predictability and consistency. In programming and real-world applications, understanding how a function operates helps in predicting the output, given any valid input. Furthermore, considering how each input relates to an output is vital when working with inverse functions.

Inverse Function Definition

Inverse functions essentially reverse the roles of inputs and outputs of a given function. If a function maps an input \(a\) to an output \(b\), its inverse will map \(b\) back to \(a\). This reversal only works perfectly if the original function is one-to-one, meaning it has an inverse.

To put it simply, consider the definition:

• If you have a function \(f\), and it maps \(a\) to \(b\) (i.e., \(f(a) = b\)), then the inverse \(f^{-1}\) does the opposite: \(f^{-1}(b) = a\).
• For functions with graphs, the line \(y = x\) acts as a mirror for the function and its inverse.

Understanding inverse functions is crucial for solving equations and analyzing changes in mathematical models. Inverse functions allow you to work backward, making them invaluable in problem-solving and theoretical mathematics.