Problem 71 - Algebra and Trigonometry

Question

A one-to-one function is given. (a) Find the inverse of the function. (b) Graph both the function and its inverse on the same screen to verify that the graphs are reflections of each other in the line \(y=x .\) $$ f(x)=2+x $$

Step-by-Step Solution

Answer

The inverse of the function is \( f^{-1}(x) = x - 2 \).

Step 1: Understand the Function

The given function is \( f(x) = 2 + x \). This is a simple linear function that maps any value of \( x \) to \( x + 2 \). To find the inverse, we need to 'reverse' this operation.

Step 2: Swap Variables

To find the inverse of a function \( f(x) \), we start by swapping \( x \) and \( y \) in the equation \( y = f(x) \). So we have: \( x = 2 + y \).

Step 3: Solve for y

Now, solve for \( y \) in terms of \( x \). Subtract 2 from both sides: \( y = x - 2 \). This is the inverse function: \( f^{-1}(x) = x - 2 \).

Step 4: Verify the Inverse Function

To confirm \( f^{-1}(x) = x - 2 \) is the correct inverse, check by composing \( f \) and \( f^{-1} \). Compute \( f(f^{-1}(x)) \) and \( f^{-1}(f(x)) \), both should return \( x \). For \( f(f^{-1}(x)) \), we have \( f(x - 2) = 2 + (x - 2) = x \). For \( f^{-1}(f(x)) \), we have \( f^{-1}(2 + x) = (2 + x) - 2 = x \). Both verify the correctness.

Step 5: Graph the Functions

Graph both \( f(x) = 2 + x \) and \( f^{-1}(x) = x - 2 \) on the same coordinate plane. These graphs should be mirror images of each other across the line \( y = x \).

Key Concepts

One-to-One Function

A one-to-one function is a special type of function in mathematics where each element of the range is paired with exactly one unique element of the domain. This means that no two different inputs will map to the same output. A one-to-one function is also sometimes called an injective function.

To determine if a function is one-to-one, you can use the horizontal line test. If any horizontal line crosses the graph of the function at most once, then the function is one-to-one. For identifying such functions:

• Every input has a unique output.
• Passed horizontal line test on graph.
• Potentially invertible with an inverse function that is also a valid function.

Since inverse functions "reverse" the effects of the original function, they only exist for one-to-one functions. In our example, the function given is linear and hence satisfies the criterion of being one-to-one.

Linear Function

Linear functions are one of the simplest types of functions characterized by their constant rate of change and graphically represented by a straight line. The general form of a linear function is written as \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

In the given exercise, the function \( f(x) = 2 + x \) can be expressed in this standard form with a slope \( m = 1 \) and y-intercept \( b = 2 \). This means the function increases at a consistent rate of 1, and it crosses the y-axis at 2. Features of linear functions include:

• Graph is a straight line.
• Has a constant slope \( m \).
• Simple arithmetic operations for finding inverses.

Linear functions are extensively studied due to their simplicity and foundational role in calculus and algebra. They're easy to invert, as shown in step-by-step solutions through simple algebraic manipulation.

Graphical Representation

Graphical representation of functions is essential to visually comprehend and verify mathematical relationships. When graphing both a function and its inverse, a critical concept to observe is their symmetry about the line \( y = x \). This line acts as a mirror, demonstrating how the function and the inverse are reflections of each other.

To graph a function and its inverse:

• Plot both the original function \( f(x) \) and the inverse \( f^{-1}(x) \).
• Ensure they reflect each other along the line \( y = x \).
• For our example, the function \( f(x) = 2 + x \) and inverse \( f^{-1}(x) = x - 2 \) will appear as mirror images.

This graphical interpretation confirms the understanding of the function's and its inverse's relationship. By practicing this method, it becomes easier to grasp the interplay between functions and their inverses, particularly for linear functions.