Problem 60 - Intermediate Algebra

Question

Find the inverse of each function. Then graph the function and its inverse on one coordinate system. Show the line of symmetry on the graph. $$ f(x)=-3 x $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Function

The given function is linear, written as \( f(x) = -3x \). To find the inverse, we first need to consider a one-to-one function that can be reversed.

Step 2: Swap and Solve for the Inverse

For the function \( y = -3x \), swap \( x \) and \( y \) to get \( x = -3y \). Solve for \( y \) by dividing both sides by -3, which gives \( y = \frac{-x}{3} \). The inverse function is \( f^{-1}(x) = \frac{-x}{3} \).

Step 3: Graph the Original Function

Graph the original function \( f(x) = -3x \). This is a straight line with a slope of -3, passing through the origin (0,0).

Step 4: Graph the Inverse Function

Graph the inverse function \( f^{-1}(x) = \frac{-x}{3} \). This is also a straight line with a slope of \(-\frac{1}{3}\), passing through the origin (0,0).

Step 5: Draw the Line of Symmetry

In the coordinate plane, draw the line of symmetry \( y = x \). This line acts as a mirror line, showing symmetry between the original function and its inverse.

Step 6: Confirm the Inverse

Check that compositions of the original and inverse functions return the inputs: \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \). For \( f(f^{-1}(x)) \), substitute the inverse function into the original: \(-3(\frac{-x}{3}) = x\). Do the same reverse: substituting the original function into the inverse gives \(\frac{-(-3x)}{3} = x\). Both give \( x \), confirming correct inversion.

Key Concepts

Linear Functions

Linear functions are a type of function that creates a straight line when graphed. They are in the form of \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. These simple expressions can model many real-world phenomena, like constant speed or uniform rates. The simplicity of a linear equation allows us to easily find the inverse.

• Slope (\( m \)): This number in the equation describes how steep the line is. In our example, the original function \( f(x) = -3x \) has a slope of -3, indicating the line steeply descends as it moves to the right.
• Y-intercept (\( b \)): The y-intercept is where the line crosses the y-axis. Here, the original function doesn't have a y-intercept (b=0) since it goes through the origin.

To find the inverse, flip \( x \) and \( y \), and solve for the new \( y \). This reversal unravels the function's operations, helping us trace how inputs and outputs are switched.

Graphing Functions

When graphing linear functions and their inverses, we illustrate how these equations translate into visual lines on a coordinate plane. This makes understanding interactions between functions clearer, especially when combined with graph features like slope and intercepts.

• Original Function: For our example, \( f(x) = -3x \), you start from the origin and plot a line descending sharply, due to the slope of -3.
• Inverse Function: The inverse, \( f^{-1}(x) = \frac{-x}{3} \), depicts a line sloping less steeply in the opposite direction, with a slope of \(-\frac{1}{3}\).

Visual tools like graphs allow us to see both functions on one plane, offering insights into their relationship. For thoroughness, include both functions on one graph to illustrate symmetry and function interaction.

Line of Symmetry

The line of symmetry for a function and its inverse is pivotal in understanding their geometric relationship. This line, \( y = x \), serves as a reflection point where the original function reflects toward its inverse.

• Symmetry: This line shows that every point on \( f(x) \) is mirrored onto \( f^{-1}(x) \), and vice versa around this line.

In our example, the symmetry line isn't just a conceptual tool; it acts as a check for accuracy. A correct graph of one function and its inverse should appear as if they mirror each other along this diagonal line. Drawing \( y = x \) on a coordinate plane with the functions enhances understanding by visualizing this concept.