Question

Consider the function \(f(x)=-x\). a. Graph \(f(x)=-x\) and explain why it is its own inverse. Also, verify that \(f(x)=-x\) is its own inverse algebraically. b. Graph other linear functions that are their own inverses. Write equations of the lines you graphed. c. Use your results from part (b) to write a general equation describing the family of linear functions that are their own inverses.

Short answer

So, the function \(f(x) = -x\) is indeed its own inverse. Its graph is a line passing through the origin with a negative slope. Other examples of such functions are \(f(x)=-2x\) or \(f(x)=-3x\) in general the function \(f(x)=ax\) for any \(a\) that is a negative real number is its own inverse.

Step-by-step solution

Graphing and Verifying for the specific function

First, graph the function \(f(x)=-x\). A graph of \(f(x)=-x\) is a straight line which passes through the origin and has a negative slope of 1. This implies the function reflects any input value about the origin, this being one demonstration of why it is its own inverse. However, we still need to verify this algebraically. If the function is its own inverse, then applying the function twice, i.e. \(f(f(x))\), should return \(x\). Checking this gives: \(f(f(x)) =f(-x) = -(-x) = x\). As this is true, it is confirmed that the function is its own inverse.

Graphing and Finding Other Self-Inverse Functions

Similarly, consider the general linear function \(f(x)=ax\). Graphing this function reveals that only those lines that pass through the origin can possibly be self-inverse, this is because self-inverse functions have to reflect every input about the origin. So, the only candidates would be functions with negative slopes. Graph functions like \(f(x)=-2x\), \(f(x)=-3x\) and so on.

Writing a General Equation

Now, writing an equation which describes all such functions, consider that these must be functions of form \(f(x)=ax\). From part a and part b, we find out that for \(f(f(x))=x\) to hold, \(a\) must be negative. Thus, the general form would be \(f(x)=ax\), where \(a<0\).

Key concepts

Understanding Linear Functions

Linear functions are the simplest forms of polynomial functions. They are generally represented by the equation of a straight line, which is typically in the form \[ y = mx + b \] where

• m is the slope, indicating the steepness of the line,
• b is the y-intercept, the point where the line crosses the y-axis.

For a function to be considered linear, each input value needs to produce an output that creates a straight line when graphed.

Linear functions are foundational in mathematics because they exhibit a constant rate of change and appear frequently in various applications. In the context of inverse functions, the linear form retains its simplicity, but specific types such as functions passing through the origin and having negative slopes prove to be of particular interest.

Algebraic Verification of Inverse Functions

When verifying if a function is its own inverse algebraically, the process involves some substitutions and simplifications. A function, when applied to itself, should yield the original input value if it is indeed its own inverse. For example, consider the function \[ f(x) = -x \].

To verify it algebraically:

• Apply the function to itself: \[ f(f(x)) = f(-x) = -(-x) = x \].

Since we obtain the original input, x, after applying the function twice, \[ f(x) = -x \] proves to be self-inverse.

This demonstrates a self-consistency check crucial for inverse verification. This algebraic process emphasizes the reflection property inherent in self-inverse functions. Such reflections usually happen around a line or point, and for linear functions like the ones discussed here, it's typically about the origin.

Exploring Self-Inverse Properties

Self-inverse properties describe functions that, when applied twice, produce the original input. For linear functions, this is somewhat limited.

Considerations about self-inverse properties generally lead us to functions where the slope is negative because:

• The line must have a symmetry around the origin.
• Reflect the input over to yield itself after iteration.

Such a property is demonstrated perfectly with \[ f(x) = -x \], where the function effectively flips the sign of the input twice to return it to its starting state.

Self-inverse functions are unique because not every function or even linear function can claim this property. It's this special symmetry and behavior showcased graphically and algebraically that paves the way for practical applications and intuitive understanding.

Graphing Functions and Their Inverses

Graphically analyzing functions, especially linear ones, gives clarity to the inverse relationship. When graphing a self-inverse function like \[ f(x) = -x \], we notice it passes through the origin with a slope of -1, showcasing a perfect reflection over both axes simultaneously.

These visual tools aid in comprehending why such a function is self-inverse. To explore other functions that exhibit self-inverse properties, graph lines like \[ f(x) = -2x \] or \[ f(x) = -3x \],as both pass through the origin and reflect their inputs similarly.

As a general form, these functions need to be embrace the format \[ f(x) = ax \],where \[ a < 0 \].

• This format ensures the line passes through the origin.
• Has an exact negative slope.

Through these graphical representations, we achieve a clearer understanding of how linear functions that are their own inverse behave and appear visually, solidifying our conceptual grasp.