Problem 54 - Algebra for College Students

Question

(a) find \(f^{-1}\) and (b) graph \(f\) and \(f^{-1}\) on the same set of axes. $$ f(x)=-3 x-3 $$

Step-by-Step Solution

Answer

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

Step 1: Understand Inverse Functions

Inverse functions essentially swap the roles of inputs and outputs. If \( y = f(x) \) and \( f^{-1} \) is its inverse, then \( f^{-1}(y) = x \). The function \( f \) and its inverse \( f^{-1} \) should satisfy \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \). We need to change \( y \) back to \( x \) in terms of the equation.

Step 2: Set Up the Equation for Inversion

The given function is \( f(x) = -3x - 3 \). First, replace \( f(x) \) with \( y \) to form the equation: \( y = -3x - 3 \).

Step 3: Swap and Solve for x

To find the inverse, swap \( x \) and \( y \). This gives us \( x = -3y - 3 \). We need to solve for \( y \) in terms of \( x \).

Step 4: Rearrange to Solve for y

Add 3 to both sides: \( x + 3 = -3y \). Then, divide both sides by -3 to isolate \( y \): \( y = -\frac{x+3}{3} \). Thus, \( f^{-1}(x) = -\frac{x+3}{3} \).

Step 5: Graph the Original and Inverse Functions

Graph \( f(x) = -3x - 3 \), which is a straight line with slope -3 and y-intercept -3. For \( f^{-1}(x) = -\frac{x+3}{3} \), graph this line with slope \(-\frac{1}{3}\) and y-intercept -1. They should reflect over the line \( y = x \).

Key Concepts

Inverse Function Calculation

Inverse functions are a fascinating concept where the output of a function becomes the input, and vice versa. To find an inverse function, you effectively "reverse" the process of the original function. Let's break down exactly how this is done using our given function, \(f(x) = -3x - 3\).\

• Swap and Solve: Start by replacing \( f(x) \) with \( y \) to get \( y = -3x - 3 \).
• Now, exchange \( x \) and \( y \) to switch their roles, resulting in \( x = -3y - 3 \).
• Solve for \( y \): add 3 to both sides to get \( x + 3 = -3y \). Now, divide both sides by -3 to isolate \( y \), giving \( y = -\frac{x+3}{3} \).
• Finally, write the inverse function as \( f^{-1}(x) = -\frac{x+3}{3} \).

If calculated correctly, applying the original and inverse functions sequentially should return the input value. Thus, \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \). This relationship further enforces the concept that an inverse function serves as a mirror to the original.

Graphing Functions

Graphing both a function and its inverse can visually clarify their relationship by seeing one as a reflection of the other over the line \( y = x \). Let's graph both \( f(x) = -3x - 3 \) and its inverse, \( f^{-1}(x) = -\frac{x+3}{3} \), to understand their graphical relationships.\

• Original Function \( f(x) \): This is a straight line with a slope of -3 and a y-intercept at -3.
• Inverse Function \( f^{-1}(x) \): This is also a straight line but with a slope of \(-\frac{1}{3}\) and y-intercept at -1.
• Reflective Line \( y = x \): Both graphs should appear mirrored across this line, highlighting the inverse relationship between the two.

This visual reflection underpins the mathematical concept of inverse functions, where plotting them reveals how closely they reverse each other's effects. By graphing, you can directly observe that every point on \( f(x) \) corresponds to a reflected point on \( f^{-1}(x) \). This clear reflection over the line \( y = x \) confirms their inverse characteristics.

Linear Transformations

Linear transformations are significantly important in understanding how functions behave under algebraic changes, like shifts, reflections, and rotations. When dealing with linear functions such as \( f(x) = -3x - 3 \), recognizing patterns of transformation can aid in mastering their graphing and manipulation.\

• Reflection: Both the original function and its inverse demonstrate reflection. The initial line reflects over the main diagonal line \( y = x \), showcasing reversal of input-output mappings.
• Shifting: The constant term in the function, such as \(-3\) in \( f(x) = -3x - 3 \), shifts the line downward on the y-axis by 3 units. In the inverse \( f^{-1}(x) = -\frac{x+3}{3} \), the term \( -1 \) indicates a similar downward y-axis shift.
• Slope Change: The slope indicates the steepness of the line; \(-3\) for \( f(x) \) and \(-\frac{1}{3} \) for \( f^{-1}(x) \) reveals how the rate of change affects transformation.

Linear transformations provide the blueprint for graphing different forms of a function. By modifying slopes and intercepts, these fundamental tweaks help in building a clear insight into how functions behave graphically and mathematically.