Problem 61 - Algebra and Trigonometry

Question

A function \(f\) is given. (a) Sketch the graph of \(f .(b)\) Use the graph of \(f\) to sketch the graph of \(f^{-1} .(c)\) Find \(f^{-1} .\) $$ f(x)=3 x-6 $$

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(x) = \frac{x+6}{3}\).

Step 1: Understand the Given Function

The function given is a linear function, i.e., a straight line: \( f(x) = 3x - 6 \). This line has a slope of 3 and a y-intercept at \(-6\).

Step 2: Sketch the Graph of f(x)

To sketch \( f(x) = 3x - 6 \), start by plotting the y-intercept (0, -6) on the Cartesian plane. Then, from this point, use the slope of 3 (which means rise 3 and run 1) to find another point, e.g., (1, -3). Connect these points to form the line.

Step 3: Concept of Inverse Function

To sketch the graph of \( f^{-1} \), we need to reflect the graph of \( f(x) \) over the line \( y = x \). The inverse function will pass through the points that are swaps of the points on \( f(x) \).

Step 4: Sketch the Graph of f^{-1}(x)

Swap the coordinates of the points on \( f(x) \). For example, if \( f(x) \) passes through (0, -6), \( f^{-1}(x) \) will pass through (-6, 0). Use another point such as (1, -3) becoming (-3, 1). Reflect the line across \( y = x \).

Step 5: Find the Inverse Function f^{-1}(x)

To find \( f^{-1}(x) \), solve the equation \( y = 3x - 6 \) for \( x \). \( y = 3x - 6 \Rightarrow y + 6 = 3x \Rightarrow x = \frac{y+6}{3} \). Thus, \( f^{-1}(x) = \frac{x+6}{3} \).

Key Concepts

Linear Functions

Linear functions are among the most basic types of mathematical functions you will encounter. Their defining characteristic is their straight-line graph. A linear function can generally be written in the form:

• \( f(x) = mx + b \)

where \( m \) represents the slope, and \( b \) represents the y-intercept. In our exercise, the function given is \( f(x) = 3x - 6 \). This equation tells us a few important things:

• The slope \( m = 3 \) indicates how steep the line is and the direction it goes. A positive slope means the line rises as it moves from left to right.
• The y-intercept \( b = -6 \) tells us where the line crosses the y-axis. The graph will pass through the point (0, -6).

By understanding these elements, we can easily sketch or interpret linear functions.

Graphing Functions

Graphing a function like \( f(x) = 3x - 6 \) involves plotting points and connecting them into a straight line. Here's how you can do it:

• Start by plotting the y-intercept at the point (0, -6). This is where the line will intersect with the y-axis.
• Use the slope of the function to find another point. The slope of 3 can be interpreted as "rise over run," meaning for every 3 units you move up, you move 1 unit to the right. From (0, -6), moving up 3 units and right 1 unit gives you the point (1, -3).
• Join these two points with a straight line, extending it in both directions to cover the graph.

This is the complete graph of the linear function \( f(x) = 3x - 6 \). These steps can be easily followed with any linear function by just updating the slope and y-intercept values as needed.

Function Reflection

Understanding inverse functions involves the concept of reflecting a graph over the line \( y = x \). The inverse function, denoted as \( f^{-1}(x) \), essentially swaps the x and y values of the original function \( f(x) \).

• To sketch an inverse, reflect points on the graph of \( f(x) \) over the line \( y = x \).
• Pretending the line \( y=x \) is a mirror, points like (0, -6) on \( f(x) \) would reflect to (-6, 0) on \( f^{-1}(x) \).
• This method works because, for an inverse function, the input and output are switched.

To find the equation of the inverse function of \( f(x) = 3x - 6 \), solve for \( x \) in terms of \( y \):

• Start with \( y = 3x - 6 \).
• Rearrange to get \( 3x = y + 6 \).
• Solve for \( x \) to get \( x = \frac{y + 6}{3} \).

Thus, the inverse function is \( f^{-1}(x) = \frac{x + 6}{3} \), completing the reflection process.