Analyzing One-to-One Functions and Their Inverses Using Graphs

Q4. (a) For each function, determine whether it is one-to-one. (b) The graph of a function is given. Draw the graph of its inverse.

Background

Topic: One-to-One Functions and Inverses

This question tests your understanding of how to determine if a function is one-to-one (injective) by analyzing its graph, and how to sketch the graph of its inverse. A function is one-to-one if every horizontal line intersects the graph at most once. The graph of the inverse function is a reflection of the original function across the line $y = x$.

Key Terms and Concepts:

• One-to-One Function: A function $f$ is one-to-one if $f(a) = f(b)$ implies $a = b$.

• Horizontal Line Test: If every horizontal line intersects the graph of $f$ at most once, then $f$ is one-to-one.

• Inverse Function: The inverse $f^{-1}$ of $f$ satisfies $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

• Graph of the Inverse: Reflect the graph of $f$ across the line $y = x$ to obtain the graph of $f^{-1}$.

Step-by-Step Guidance

1. For each graph, apply the horizontal line test to determine if the function is one-to-one. Imagine or draw horizontal lines and see if any line crosses the graph more than once.

   

2. If a graph passes the horizontal line test (no horizontal line crosses more than once), the function is one-to-one. If any horizontal line crosses more than once, it is not one-to-one.

3. For part (b), to draw the graph of the inverse, reflect each point of the original function across the line $y = x$. For example, a point $(a, b)$ on the original graph becomes $(b, a)$ on the inverse.

   

4. Check that the reflected graph is still a function (passes the vertical line test) and matches the expected behavior of an inverse.

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