Problem 46 - Intermediate Algebra

Question

(a) find the inverse of the given function, and (b) graph the given function and its inverse on the same set of axes. (Objective 4) $$f(x)=2 x+2$$

Step-by-Step Solution

Answer

Inverse: \( f^{-1}(x) = \frac{x - 2}{2} \); Graph both functions and their symmetry about the line \( y = x \).

Step 1: Understand the Problem

We are given the function \( f(x) = 2x + 2 \) and are asked to find its inverse and graph both the function and its inverse on the same axes.

Step 2: Swap Variables

To find the inverse of a function, start by swapping \( x \) and \( y \). Since \( y = f(x) = 2x + 2 \), we write \( x = 2y + 2 \).

Step 3: Solve for y

Now, solve the equation for \( y \) to find the inverse function. Start by subtracting 2 from both sides: \( x - 2 = 2y \). Then, divide both sides by 2: \( y = \frac{x - 2}{2} \). Thus, the inverse function is \( f^{-1}(x) = \frac{x - 2}{2} \).

Step 4: Identify the Domain and Range

The domain of \( f(x) = 2x + 2 \) is all real numbers, \( \mathbb{R} \). The inverse function has the same domain and range, as it is a line. Similarly, the range of both \( f(x) \) and \( f^{-1}(x) \) is all real numbers.

Step 5: Graph the Functions

To graph \( f(x) = 2x + 2 \), find two points; for example, use \( x = 0 \) to get \( y = 2 \), and \( x = 1 \) to get \( y = 4 \). Plot these points and draw a straight line through them. For \( f^{-1}(x) = \frac{x - 2}{2} \), use points like \( x = 2 \) to get \( y = 0 \), and \( x = 4 \) to get \( y = 1 \). Plot these and draw the line. Both graphs should be symmetric about the line \( y = x \).

Step 6: Verification of Results

Verify that \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \). For instance, \( f(f^{-1}(x)) = 2\left(\frac{x - 2}{2}\right) + 2 = x \) confirms correctness. Similarly, verify \( f^{-1}(f(x)) = \frac{(2x + 2) - 2}{2} = x \). This shows that both functions are indeed inverses.

Key Concepts

Graphing Functions

Graphing functions helps us visualize their behavior and how they transform. For the given function, \( f(x) = 2x + 2 \), which is a linear function, graphing it starts with selecting points that satisfy the equation. Consider \( x = 0 \) and \( x = 1 \) to find two points, \( (0, 2) \) and \( (1, 4) \). These points are typically sufficient for a linear function since the graph will be a straight line. Once plotted on a coordinate plane, draw a line through these points. This graph shows that as \( x \) increases by 1, \( y \) increases by 2, which affirms the slope of the line is 2. The y-intercept, \( +2 \), means the line crosses the y-axis at \( y = 2 \).Now for the inverse function, \( f^{-1}(x) = \frac{x - 2}{2} \), graphing follows a similar process. Using \( x = 2 \) to get \( y = 0 \) and \( x = 4 \) for \( y = 1 \), plot these points and draw a line. The line of \( f^{-1} \) will be symmetrical to the line of \( f \) when reflected over \( y = x \). This reflects the concept that inverse functions undo each other's operations.

Domain and Range

The domain and range are key features of a function. For the function \( f(x) = 2x + 2 \), the domain includes all real numbers \( \mathbb{R} \) since a linear function continues infinitely in both positive and negative directions on the x-axis without any restrictions. Similarly, the range, the set of all possible outputs \( y \), also covers all real numbers. This linear function can produce any real number as \( y \), which is evident since there is no limit to the value of \( x \).For the inverse function \( f^{-1}(x) = \frac{x - 2}{2} \), the domain and range swap compared to the original function. However, since both functions are linear and extend indefinitely, the domain and range for the inverse function remain \( \mathbb{R} \) as well. This characteristic of linear functions ensures that every real number input provides a real number output and vice versa.

Function Symmetry

Symmetry in functions, particularly with inverses, shows a fascinating reflection property. For \( f(x) = 2x + 2 \) and its inverse \( f^{-1}(x) = \frac{x - 2}{2} \), this symmetry occurs about the line \( y = x \). This line, often referred to as the line of symmetry for inverses, implies that each point \((a, b)\) on \( f(x) \) corresponds to a point \((b, a)\) on \( f^{-1}(x) \).The symmetry about \( y = x \) is crucial, as it graphically illustrates how inverses work. The graphs of \( f \) and \( f^{-1} \) should ideally fold onto each other along this line, showcasing that applying either function moves along the path of the other. This visual tool aids in verifying that two functions are indeed inverse to one another, and helps in understanding the fundamental relationship between a function and its inverse.