Problem 38 - 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. $$f(s)=2 s-\frac{9}{5}$$

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(s) = \frac{s + \frac{9}{5}}{2}\). When graphed, the original function and its inverse are mirror images across the line \(y = x\).

Step 1: Find Inverse of the Function

To find the inverse of the function \(f(s) = 2s - \frac{9}{5}\), swap \(s\) and \(f\) to get \(s = 2f - \frac{9}{5}\). Then solve for \(f\), which yields \[f_{inverse}(s) = \frac{s + \frac{9}{5}}{2}\]

Step 2: Graph the Original Function

Graph the original function \(f(s) = 2s - \frac{9}{5}\) on the Cartesian plane. This is a linear function with a slope of 2 and a Y-intercept of \(-\frac{9}{5}\). Start at point (0, \(-\frac{9}{5}\)) on the Y-axis and slope upwards by 2 units for every 1 unit moved to the right.

Step 3: Graph the Inverse Function

Next, plot the inverse function \[f_{inverse}(s) = \frac{s + \frac{9}{5}}{2}\] on the same set of axes. This function also is linear with a slope of \(\frac{1}{2}\) and a Y-intercept of \(\frac{9}{10}\). Start from point (0, \(\frac{9}{10}\)) and slope upwards by 0.5 units for every 1 unit moved to the right. The graphs of the function and its inverse are reflections of each other across the line \(y = x\).

Key Concepts

Graphing Functions

Graphing functions helps us visualize and understand the behavior of functions on the Cartesian plane. It involved plotting points that satisfy the given function's equation. For example, when graphing a function like \( f(s) = 2s - \frac{9}{5} \), knowing the equation allows you to find key characteristics, such as the slope and the intercepts, which guide you in sketching the graph.

To graph a linear function, you start by identifying crucial elements such as:

• Y-Intercept: This is where the graph crosses the Y-axis. For \( f(s) = 2s - \frac{9}{5} \), the Y-intercept is \(-\frac{9}{5}\).
• Slope: This indicates the steepness or incline of the line. Here, it's 2, meaning the line rises 2 units for every 1 unit moved to the right.

Begin plotting from the Y-intercept and use the slope to find additional points. Once plotted, draw a straight line through these points. Don't forget, when you graph an inverse, which inverts the independent and dependent variables, both graphs should be mirrored across the line \( y = x \).

Linear Functions

Linear functions are the simplest type of function characterized by a constant rate of change and a straight-line graph. They generally take the form \( y = mx + b \), often referred to as the slope-intercept form, and represent a straight line.

With linear functions, the slope \( m \) tells us how steep the line is. If \( m > 0 \), the line ascends; if \( m < 0 \), it descends. The Y-intercept \( b \) shows where the line crosses the Y-axis.

Inverse functions of linear functions also form straight lines. An inverse function is essentially the reflection of the original function across the line \( y = x \). Given our function \( f(s) = 2s - \frac{9}{5} \), its inverse, \( f_{inverse}(s) = \frac{s + \frac{9}{5}}{2} \), will provide insights into points that reverse the order of the original function's input and output.

Slope-Intercept Form

The slope-intercept form of a linear equation \( y = mx + b \) is incredibly useful for graphing linear functions quickly. It clearly lays out the slope \( m \) and the Y-intercept \( b \).

Understanding the slope is crucial. It's the ratio of the rise (change in Y) over the run (change in X). For our function, the slope is 2, and thus, for every 1 unit move to the right, the function values rise by 2 units. Meanwhile, the graph starts at the Y-intercept, \(-\frac{9}{5}\), the point where the line intersects the Y-axis.


In contrast, for the inverse function \( f_{inverse}(s) = \frac{s + \frac{9}{5}}{2} \), the slope is \( \frac{1}{2} \), indicating a gentler upward incline than the original function. Its Y-intercept is \( \frac{9}{10} \). This variation in slope and intercept between the function and its inverse distinctly illustrate their reflective relationship across the line \( y = x \).