Problem 3 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = -x \); it is the same as the original function.

Step 1: Understand the Function

The function given is \( f(x) = -x \). This is a linear function that reflects across the origin. In its basic form, it's an identity function that reflects each point to its opposite in the Cartesian plane.

Step 2: Swapping Variables

To find the inverse, we start by swapping the function roles of \( x \) and \( y \). Thus, we have \( x = -y \).

Step 3: Solving for y

Next, solve the equation from Step 2 for \( y \). Starting with the equation \( x = -y \), we can rearrange this into \( y = -x \). Thus, the inverse function is \( f^{-1}(x) = -x \).

Step 4: Graphing Original and Inverse Functions

Both the original function \( f(x) = -x \) and its inverse \( f^{-1}(x) = -x \) are the same line, \( y = -x \). When graphed, this line passes through the origin and forms a 45-degree line going from the top-left to the bottom-right of the graph.

Key Concepts

Linear Functions

Linear functions are fundamental building blocks in mathematics and appear in many real-world applications. A linear function is typically written in the form \( f(x) = ax + b \), where \( a \) and \( b \) are constants. This expression represents a straight line when graphed on the Cartesian plane.
In the case of the given function \( f(x) = -x \), it is a special type of linear function with a slope of \(-1\) and a y-intercept of 0. The slope tells us that the line is going downward at a consistent rate.
Linear functions are straightforward:

• They have constant rates of change.
• They generate straight lines when graphed.
• Simple to work with, making them ideal for problems requiring inversion or transformation.

Understanding linear functions is critical as they lay the foundation for more complex functions and their inverses.

Graphing Functions

Graphing functions is an essential skill that helps visualize how functions behave. It translates the algebraic equation into a visual format.
For the function \( f(x) = -x \), its graph is a straight line, as mentioned earlier. Here's how you can graph it:

• Identify points: select a few values for \( x \) and compute corresponding \( y \) values. For example, if \( x = 1 \), then \( y = -1 \), and if \( x = 2 \), then \( y = -2 \).
• Plot these points on graph paper or using graphing software.
• Draw a line through these points, making sure it extends across your graph.

Graphing is not only about plotting points. It involves understanding the nature of the function, like symmetry and intercepts, and observing these characteristics visually.
In the inverse function \( f^{-1}(x) = -x \), the graph is identical because it perfectly reflects over the line \( y = x \), showing the concept of invertibility is retained.

Reflection Across the Origin

Reflection across the origin is a specific type of symmetry found in functions. Sometimes known as origin symmetry, it occurs when every point \((x, y)\) on the function has a corresponding point \((-x, -y)\). This property is evident in the function \( f(x) = -x \).
The function \( f(x) = -x \) reflects every input-output pair over the origin, resulting in the same line for both the original and inverted functions.
Key characteristics of reflection across the origin include:

• The graph appears identical after a 180-degree rotation around the origin.
• Functions that reflect across the origin are _odd functions_.
• The behavior signifies that the function is its own inverse.

Origin symmetry makes identifying inverse functions intuitive and shows the geometric relationship between a function and its inverse.