Determine whether each pair of functions graphed are inverses.

Verified step by step guidance

1. Step 1: Understand the concept of inverse functions. Two functions are inverses if their graphs are reflections of each other across the line \(y = x\).
2. Step 2: Identify the two functions graphed. In the image, the orange and blue lines represent the two functions, and the green dashed line is the line \(y = x\).
3. Step 3: Check if the graphs of the two functions are symmetric with respect to the line \(y = x\). This means that for every point \((a, b)\) on one function, there should be a corresponding point \((b, a)\) on the other function.
4. Step 4: Observe the graph carefully. The orange and blue lines appear to be reflections of each other across the green dashed line \(y = x\), indicating that they are inverses.
5. Step 5: Conclude that since the two functions are symmetric about the line \(y = x\), the pair of functions graphed are indeed inverses of each other.

Key Concepts

Inverse Functions

Inverse functions reverse the effect of each other, meaning if f(x) maps x to y, then its inverse f⁻¹(x) maps y back to x. Graphically, two functions are inverses if their graphs are reflections of each other across the line y = x.

Line of Reflection (y = x)

The line y = x acts as the mirror line for inverse functions. If one function's graph is reflected over this line, it should coincide with the graph of its inverse. This line helps visually verify if two functions are inverses.

Graphical Verification of Inverses

To determine if two functions are inverses using their graphs, check if each point (a, b) on one graph corresponds to a point (b, a) on the other. This means the graphs are symmetric about the line y = x, confirming the inverse relationship.