In Exercises 7–10, determine from its graph if the function is one-to-one.
f(x) = 3 - x, x < 0
= 3, x ≥ 0

Verified step by step guidance

1. Understand the definition of a one-to-one function: a function is one-to-one if and only if each output value corresponds to exactly one input value. In other words, no horizontal line intersects the graph more than once.
2. Analyze the given piecewise function: \( f(x) = \begin{cases} 3 - x, & x < 0 \\ 3, & x \geq 0 \end{cases} \). For \( x < 0 \), the function is a line with slope \(-1\), and for \( x \geq 0 \), the function is constant at 3.
3. Consider the graph of the first part \( 3 - x \) for \( x < 0 \). This is a decreasing linear function, so it is one-to-one on this interval.
4. Look at the second part where \( f(x) = 3 \) for \( x \geq 0 \). This is a horizontal line, meaning all inputs \( x \geq 0 \) map to the same output 3, which violates the one-to-one condition on this interval.
5. Check if any output value is repeated for different inputs across the two pieces. Since \( f(0) = 3 \) and also \( f(x) = 3 \) for all \( x \geq 0 \), and the first piece approaches values near 3 as \( x \to 0^- \), the function is not one-to-one overall.

Key Concepts

One-to-One Function

A function is one-to-one (injective) if each output corresponds to exactly one input. This means no two different inputs produce the same output. Checking if a function is one-to-one helps determine if it has an inverse function.

Piecewise Functions

Piecewise functions are defined by different expressions over different parts of their domain. Understanding how each piece behaves separately and together is essential to analyze properties like continuity and injectivity.

Graphical Test for One-to-One Functions

The horizontal line test is used to determine if a function is one-to-one by checking if any horizontal line intersects the graph more than once. If it does, the function is not one-to-one.