Problem 41 - Precalculus : Building Concepts and Connections

Question

Find the inverse of the given function. Then graph the given function and its inverse on the same set of axes. $$f(x)=\frac{1}{2} x-4$$

Step-by-Step Solution

Answer

The inverse of the function \( f(x) = \frac{1}{2}x - 4 \) is \( f^{-1}(x) = 2x + 8 \). Both the original function \( f(x) \) and its inverse \( f^{-1}(x) \) are linear functions and create straight lines. They are symmetric over the line \( y = x \).

Step 1: Find the inverse

To find the inverse of a function, we swap the roles of \( x \) and \( y \). Starting with \( y = \frac{1}{2}x - 4 \), this becomes \( x = \frac{1}{2}y - 4 \). Solving for \( y \), the inverse function is \( y = 2x + 8 \). Hence, the inverse of \( f \) is \( f^{-1}(x) = 2x + 8 \).

Step 2: Plot the original function

The original function \( f(x) = \frac{1}{2}x - 4 \) is a straight line with a slope of \( \frac{1}{2} \) and a y-intercept of -4. Plot a point at (-4, 0) and draw a line with a slope \( \frac{1}{2} \).

Step 3: Plot the inverse function

Next, we plot the inverse function \( f^{-1}(x) = 2x + 8 \) on the same set of axes. It is also a straight line, but with a slope of 2 and a y-intercept of 8. Plot a point at (8, 0) and draw a line with a slope of 2.

Step 4: Reflect Original Function over y=x line

Since an inverse function is a reflection of the original function over the line \( y = x \), it is noticeable that if we plot the line \( y = x \), the function \( f(x) = \frac{1}{2}x - 4 \) and its inverse \( f^{-1}(x) = 2x + 8 \) are symmetric with respect to this line.

Key Concepts

Linear Functions

Linear functions are one of the most basic yet important types of functions that you will encounter in mathematics. They describe a relationship where the change in the dependent variable is consistent with the change in the independent variable. In simple terms, it creates a straight line when plotted on a graph.

Consider the linear function given in the exercise, which is \( f(x) = \frac{1}{2}x - 4 \). Here, the function is expressed in the form \( y = mx + b \), where \( m \) represents the slope, and \( b \) is the y-intercept. The slope \( \frac{1}{2} \) indicates that for every unit increase in \( x \), \( y \) increases by half a unit. The negative y-intercept \(-4\) shows that the line crosses the y-axis at \( (0, -4) \).

Linear functions are powerful due to their predictability and simplicity. They're often used in real-world applications where relationships are consistent, such as calculating speed given time and distance. Understanding linear functions is crucial as it lays the foundation for more complex mathematical concepts.

Graphing Functions

Graphing functions is an essential skill that allows us to visualize mathematical relationships. It involves plotting the function's output values (y-values) against its input values (x-values) to reveal the shape of the function on a coordinate plane.

For the original function \( f(x) = \frac{1}{2}x - 4 \), graphing involves:

• Identifying the y-intercept, \(-4\), and plotting the point \((0, -4)\).
• Using the slope \( \frac{1}{2} \) to determine another point by measuring up 1 unit and right 2 units from the y-intercept.
• Drawing a line through these points to extend infinitely in both directions.

When graphing its inverse, \( f^{-1}(x) = 2x + 8 \), a similar process is followed:

• Plot the y-intercept at point \((0, 8)\).
• Use the slope, \( 2 \), to find another point by going up 2 units and right 1 unit from the y-intercept.
• Connect these points with a straight line.

Graphing gives a clear representation of the behavior and relationships between the original and inverse function, allowing for better understanding and analysis.

Reflection Over the Line y=x

The concept of reflection over the line \( y = x \) is pivotal when dealing with functions and their inverses. The line \( y = x \) acts as a mirror that shows us the symmetry between a function and its inverse.

When we find the inverse of a function, every point \((x, y)\) on the original function corresponds to a point \((y, x)\) on the inverse. This reflection property means that if you plot the line \( y = x \) on the same graph, both the original function \( f(x) = \frac{1}{2}x - 4 \) and its inverse \( f^{-1}(x) = 2x + 8 \) will intersect this line in such a way that they are mirror images of each other.

This quality is incredibly useful because if you can visually verify that the function and its inverse are reflections over \( y = x \), it confirms the accuracy of your inverse. This principle not only helps in understanding the nature of inverses but also aids in recognizing symmetrical patterns in graphs, which is a valuable skill in mathematics.