Subjects algebra

Inverse Functions E1D119

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1. **Problem Statement:** Determine which pairs of graphed functions are inverses of each other. 2. **Key Concept:** Two functions $f$ and $g$ are inverses if and only if their graphs are symmetric about the line $y=x$. This means reflecting one graph over the line $y=x$ produces the other. 3. **Analysis of Each Pair:** - Pair 1: Two linear functions, one with positive slope through the origin and the other with negative slope, appear as reflections over $y=x$. This suggests they are inverses. - Pair 2: A curve starting in the top-left quadrant bending down to the right and a line $y=x$. The line $y=x$ is the identity function, so the other curve would have to be its own inverse. The graph does not show symmetry about $y=x$ for the pair, so not inverses. - Pair 3: An exponential growth curve and its inverse decreasing curve reflected about $y=x$. This is a classic inverse pair (e.g., $y=2^x$ and $y=\log_2 x$). They are inverses. - Pair 4: A cubic function and its reflection over $y=x$. Cubic functions are one-to-one and their inverses are also cubic-like. The reflection confirms they are inverses. - Pair 5: Two linear lines, one positive slope and one negative slope, but not symmetric about $y=x$. Not inverses. - Pair 6: An increasing curve and a line segment with no clear symmetry about $y=x$. Not inverses. 4. **Final Answer:** The pairs that are inverses are **1, 3, and 4**.