Problem 20 - Algebra 2

Question

Find the inverse of each function. Then graph the function and its inverse. $$ f(x)=3 x+3 $$

Step-by-Step Solution

Answer

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

Step 1: Write Down the Function

The given function is \( f(x) = 3x + 3 \). Our goal is to find the inverse of this function.

Step 2: Replace f(x) with y

Set \( y = f(x) \), so \( y = 3x + 3 \). This will make it easier to work with the equation for swapping variables to find the inverse.

Step 3: Swap x and y

To find the inverse, interchange \( x \) and \( y \). This gives us the equation \( x = 3y + 3 \).

Step 4: Solve for y

Rearrange the equation \( x = 3y + 3 \) to solve for \( y \). Subtract 3 from both sides to get \( x - 3 = 3y \). Then, divide both sides by 3, resulting in \( y = \frac{x - 3}{3} \).

Step 5: Substitute Back f^{-1}(x)

The expression for the inverse function is \( f^{-1}(x) = \frac{x - 3}{3} \).

Step 6: Graph the Function and Its Inverse

The graph of \( f(x) = 3x + 3 \) is a straight line with a slope of 3 and a y-intercept at 3. The graph of the inverse function, \( f^{-1}(x) = \frac{x - 3}{3} \), is also a straight line with a slope of \( \frac{1}{3} \) and a y-intercept at -1. Graph both lines on the same axes to see the symmetry about the line \( y = x \).

Key Concepts

Linear Functions

Linear functions are fundamental in algebra and mathematics, forming the building blocks for understanding more complex functions and their behaviors. A linear function is one that can be expressed in the form:\[ f(x) = mx + b \]where:

• \( m \) represents the slope of the line, which indicates the steepness or incline. The larger the value of \( m \), the steeper the line.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

A key property of linear functions is that they form a straight line when graphed. In our example, the function \( f(x) = 3x + 3 \) demonstrates a slope of 3 and a y-intercept of 3. This means it rises three units vertically for every unit it moves horizontally.Linear functions are determined by their consistent rate of change. They are straightforward to calculate and predict, which is why they serve as the foundational functions in many fields, including economics, physics, and everyday problem solving.

Graphing Functions

Graphing functions like \( f(x) = 3x + 3 \) and its inverse helps provide a visual understanding of the relationship between variables. To graph a linear function:

• Identify the slope \( m \) and the y-intercept \( b \) in the equation \( f(x) = mx + b \).
• Start by plotting the y-intercept on the graph.
• Use the slope to determine another point on the graph, starting from the y-intercept. For example, with a slope of 3, move 3 units up for every 1 unit to the right.
• Draw a straight line through the plotted points.

When graphing both a function and its inverse, it’s insightful to see their symmetry visually. The inverse of \( f(x) \) can be found graphically by reflecting the original curve across the line \( y = x \). This reflection demonstrates that for every point \((a, b)\) on the function, there is a corresponding point \((b, a)\) on the inverse.

Mathematical Solutions

Solving for the inverse of a function involves several straightforward mathematical steps that reveal the relationship between inputs and outputs. Here's how you find an inverse:

• Start with the given function, here \( f(x) = 3x + 3 \).
• Set \( f(x) = y \) to work with a single variable: \( y = 3x + 3 \).
• Switch the roles of \( x \) and \( y \) to find the inverse: \( x = 3y + 3 \).
• Solve the resulting equation for \( y \), leading to \( y = \frac{x - 3}{3} \).

Finding the inverse provides a deeper understanding of the original function's behavior. It allows you to uncover the input that would result in a given output. In practical applications, this can help solve real-world problems where you need to "reverse" an operation or process. Understanding these steps equips you with tools to handle various mathematical challenges efficiently.