Question

Explain how to determine whether the inverse of a function is also a function.

Short answer

To determine whether the inverse of a function is also a function, apply the Horizontal Line Test to the original function. If the function passes the test (i.e., no horizontal line intersects the graph of the function more than once), the inverse of the function is also a function. If it does not pass the test, the inverse is not a function.

Step-by-step solution

Understand the function and its inverse

First, identify the given function and its inverse. You need to be clear about what the original function is and what the inverse would look like. Remember, the inverse of a function 'f' is denoted by \(f^{-1}(x)\). The inverse function \(f^{-1}(x)\) reverses the job of the function.

Apply the Horizontal Line Test

The second step is to apply the Horizontal Line Test (HLT) to the given function. Draw horizontal lines through the function's graph. If any horizontal line intersects the graph of the function more than once, then the function does not pass the horizontal line test and therefore does not have an inverse that is a function.

Evaluate the result

Evaluate the results of the Horizontal Line Test to determine whether the inverse of a function exists. If the function passes the HLT, the inverse of the function is also a function. If it does not pass HLT, it means the inverse of the function is not a function.

Key concepts

Understanding the Horizontal Line Test

When we talk about the Horizontal Line Test (HLT), we're referring to a visual technique used to determine if a function has an inverse that is also a function. Imagine drawing horizontal lines across the graph of the original function. If any of these lines touch the graph at more than one point, the function fails the test. This failure means the function is not one-to-one, an essential characteristic for a function to have an inverse that is also a function.

The one-to-one property implies that for every output of the function, there is only one corresponding input. If the function passes this test, congrats! You've got a function that can be flipped over the line y = x to produce an inverse that still meets the definition of a function, meaning each input of the inverse corresponds to one and only one output.

The Basics of Function Evaluation

Moving on to the concept of function evaluation, this is the process where we 'plug in' a particular value for the variable in the function's formula and 'churn out' an output. It's the bread and butter of working with any mathematical function. For the original function, noted as f(x), and its inverse, f^-1(x), understanding how to evaluate both appropriately is crucial.

For example, if f(x) = 2x + 3, evaluating the function when x = 4 would give us f(4) = 2(4) + 3 = 11. Likewise, if we know the inverse function, we can work backward. If we have f^-1(x), and we know that f^-1(11) should give us the original x value we started with, in this case, 4.

Inverse Function Properties

Lastly, let's touch on the inverse function properties. Inverse functions essentially 'undo' what the original function did. If you think of a function as a process, like following a recipe, the inverse is like having the dish and figuring out the recipe used. One key property is that the domain of the original function becomes the range of the inverse, and vice versa.

Another interesting property is that the graphs of a function and its inverse are symmetrical along the line y = x. Because of this symmetry, f(f^-1(x)) = x and f^-1(f(x)) = x, assuming that inverse function exists. Always remember that for a function to have an inverse that is also a function, it must be both one-to-one and onto, which means it covers the entire range of possible outputs. Properties like these are what make inverse functions a fascinating area to explore in mathematics.