Determine whether each pair of functions graphed are inverses.

Verified step by step guidance

1. Step 1: Understand that two functions are inverses if their graphs are reflections of each other across the line \(y = x\). This means every point \((a, b)\) on one function corresponds to a point \((b, a)\) on the other function.
2. Step 2: Observe the given graph, which shows two functions (one in orange and one in blue) and the line \(y = x\) (green dashed line). The line \(y = x\) acts as the mirror line for checking inverses.
3. Step 3: Check if the orange and blue graphs are symmetric with respect to the line \(y = x\). This means visually verifying if the blue graph is the reflection of the orange graph across the green dashed line.
4. Step 4: Notice that the orange graph is on the right side of the \(y\)-axis and the blue graph is on the left side, and they appear to be mirror images across the line \(y = x\). This suggests they could be inverses.
5. Step 5: Conclude that since the two graphs are reflections of each other across the line \(y = x\), the pair of functions graphed are inverses of each other.

Key Concepts

Inverse Functions

Inverse functions reverse each other's operations, 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 reflecting one function's graph over the line y = x results in the other function's graph.

Line of Symmetry y = x

The line y = x acts as a mirror line for inverse functions. If two functions are inverses, their graphs are symmetric with respect to this line. Checking if one graph is the reflection of the other across y = x helps determine if they are inverses.

Graphical Verification of Inverses

To verify if two functions are inverses using their graphs, reflect one graph over the line y = x and see if it coincides with the other. This visual method provides an intuitive way to confirm inverse relationships without algebraic calculations.