Problem 53 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

In what line(s) is the graph of \(y=\frac{1}{x}\) symmetric? What does this symmetry tell you about the inverse of the function \(f(x)=\frac{1}{x}\) ?

Step-by-Step Solution

Answer

The graph of the function \(y=\frac{1}{x}\) is symmetric about the origin. This means that the function and its inverse are the same since inverse functions are the reflection of the original function in the line \(y=x with respect to the origin. Thus, the inverse function \(f^-1(x)=y=\frac{1}{x}\) is also symmetric about the origin.

Step 1: Identify the Symmetry Line

In the graph of the function \(y=\frac{1}{x}\), the line of symmetry is the origin or the point (0,0). This means the function is symmetric about the origin.

Step 2: Decipher the Implication for Inverse

The graph of \(y=\frac{1}{x}\) being symmetric about the origin would imply that its inverse function, which is \(y = \frac{1}{x}\) as well (since this is a reciprocal function), will also be symmetric about the origin. This means that the function and its inverse are the same function.

Step 3: Verify the Inverse Symmetry

Just to verify, a sketch of the function and its inverse, would show that the function \(f(x) = \frac{1}{x}\) and its inverse are indeed the same, providing further evidence that they are symmetric about the origin.

Key Concepts

Inverse Functions

An inverse function essentially reverses the effect of the original function. If you take a function \( f(x) \) and its inverse \( f^{-1}(x) \), applying the inverse function to \( f(x) \) brings you back to where you started, such that \( f^{-1}(f(x)) = x \).
For the function \( f(x) = \frac{1}{x} \), the unique part is that its inverse is itself. This means the operation of taking the reciprocal function doesn’t change the function—it remains \( f(x) = \frac{1}{x} \).

• This property is not common, as most functions differ from their inverses.
• The fact that this function is its own inverse is linked to its specific symmetry.

This self-inversive property tells us a lot about how the function interacts with other functions and how it maintains balance around the coordinate axes.

Reciprocal Function

The reciprocal function is defined as \( y = \frac{1}{x} \). It plays a crucial role in mathematical transformations and analysis.
This function is fascinating because:

• It has vertical and horizontal asymptotes at \( x = 0 \) and \( y = 0 \).
• It isn't defined at \( x = 0 \), making the x-axis a crucial boundary.
• It's symmetrical about the line \( y = x \) when viewed in a broader relational sense involving inverses.

In essence, reciprocal functions transform the idea of multiplication into a spatial relationship on the graph, showing divisional relationships across the axes.
Understanding reciprocal functions is essential for delving deeper into rational functions and behaviors.

Graph Symmetry

Graph symmetry refers to the balanced layout of a graph around a specific line or point. For the function \( y = \frac{1}{x} \), the graph is symmetric about the origin, meaning it looks the same in all quadrants if rotated 180 degrees around this point. This symmetry tells us several things:

• Every positive output has a corresponding negative counterpart.
• The rotation reveals the unchanged nature of the function with respect to its inverse.
• It gives insight into why \( y = \frac{1}{x} \) is its own inverse, maintaining a perfect mirror relationship across the axes.

Observing graph symmetry is crucial for understanding the behavior and relationships within functions. It allows us to predict and verify inverses and understand the inherent balance in mathematical expressions.