Problem 8 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the inverse function of \(f\) informally. Verify that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x.\) $$f(x)=\frac{1}{3} x$$

Step-by-Step Solution

Answer

The inverse function of \(f(x) = \frac{1}{3}x\) is \(f^{-1}(x) = 3x\). The verifications \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\) are also correct.

Step 1: Find the Inverse Function

Begin by replacing \(f(x)\) with \(y\), so the function becomes \(y = \frac{1}{3}x\). Then, swap \(x\) and \(y\) to solve for the new \(y\), which will be the inverse function. Doing so, we get \(x = \frac{1}{3}y\). Multiply both sides by 3 to solve for \(y\), and the result is that the inverse function \(f^{-1}(x)\) is \(f^{-1}(x) = 3x.\)

Step 2: Verify \(f(f^{-1}(x)) = x\)

Substitute \(f^{-1}(x)\) into \(f(x)\). This means wherever there is an \(x\) in \(f(x)\), replace it with \(f^{-1}(x)\). So, \(f(f^{-1}(x)) = \frac{1}{3}*(3x) = x\). This shows that \(f(f^{-1}(x)) = x\), proving the first verification correct.

Step 3: Verify \(f^{-1}(f(x)) = x\)

Substitute \(f(x)\) into \(f^{-1}(x)\). This means wherever there is an \(x\) in \(f^{-1}(x)\), replace it with \(f(x)\). So, \(f^{-1}(f(x)) = 3 * (\frac{1}{3}x) = x\). This shows that \(f^{-1}(f(x)) = x\), proving the second verification correct.

Key Concepts

Algebraic Functions

Algebraic functions are the bread and butter of high school algebra. These are expressions that relate two variables, typically denoted as 'x' and 'y,' through the use of algebraic operations like addition, subtraction, multiplication, division, and exponentiation with rational exponents. An example of a simple algebraic function is the linear function, which has the form

\( f(x) = mx + b \), where \( m \) and \( b \) are constants. The linear function given in the exercise, \( f(x) = \frac{1}{3}x \), is a special case where the constant \( b \) is zero, often termed a proportional linear function. It maps every x-value to one-third of that value on the y-axis. Algebraic functions can be graphed on coordinate planes, and their behaviors—such as increasing, decreasing, or remaining constant—can be analyzed to understand the relationships they express.

Function Verification

Function verification is an essential process in algebra that confirms whether one function is the inverse of another. Suppose we have a function \( f(x) \) and its presumed inverse \( f^{-1}(x) \). To verify the inverse relationship, we check two critical conditions:

• \( f(f^{-1}(x)) = x \)
• \( f^{-1}(f(x)) = x \)

These conditions must hold true for all x-values in the domain of \( f(x) \). The first condition asserts that applying the function on its inverse returns the original x-value. The second condition ensures that applying the inverse on the function also yields the original x-value. Successfully confirming both conditions tells us that the functions are indeed inverses of each other, meaning they 'undo' each other's actions.

Solving for Y

When working with functions, 'solving for y' means expressing the variable y explicitly in terms of the other variable, usually x. In the context of finding an inverse function, swapping the roles of x and y is integral. The process starts by replacing the function notation \( f(x) \) with 'y' and then solving the resulting equation for y in terms of x.

For example, starting with the equation \( y = \frac{1}{3}x \) from the exercise, after swapping to \( x = \frac{1}{3}y \), we solve for y by isolating it on one side of the equation. Multiplying both sides by 3, we get \( y = 3x \), which represents the inverse function in terms of x. This algebraic manipulation is foundational in understanding how to extract one variable from a function in terms of another, facilitating deeper comprehension of function behavior and relationships.

Inverse of a Linear Function

The inverse of a linear function is found by exchanging the roles of the 'input' and 'output' variables, which, in the case of linear functions, results in another linear function. To illuminate, let's recap the example from our textbook exercise. Given the linear function \( f(x) = \frac{1}{3}x \), to find its inverse, we swapped x and y to get \( x = \frac{1}{3}y \), and then we solved for the new y, which represents the inverse function.

In this instance, by multiplying both sides by 3, we obtained \( y = 3x \), and so our inverse function is \( f^{-1}(x) = 3x \). It's important to recognize that for any linear function that is not a vertical or horizontal line (except when the slope is zero), there will always exist a linear inverse. When graphed, the function and its inverse will be symmetrical across the line \( y = x \), a fact that can serve as a quick visual check for the correctness of an inverse function.