Question

1. error analysis: liz wrote the following piecewise-defined function:

$f(x)=\

$$\begin{cases}x - 3, & x \\leq -3 \\\\ -2x - 4, & x \\geq -3\\end{cases}$$

$
what is the error that liz made?

Explanation:

A valid piecewise function must have non-overlapping, well-defined domains for each piece, except at points where the function values match. For $x=-3$, calculate both pieces:

1. For $x=-3$ in $x-3$: $-3 - 3 = -6$
2. For $x=-3$ in $-2x-4$: $-2(-3) - 4 = 6 - 4 = 2$

The two pieces assign different values to $x=-3$, and the domain includes $x=-3$ in both pieces, creating a conflict where the function has two outputs for a single input, violating the definition of a function.

Answer:

Liz defined the piecewise function with overlapping domain $x=-3$ for both pieces, and the two pieces produce different values at $x=-3$, meaning the function is not well-defined (a single input $x=-3$ maps to two different outputs, which violates the definition of a function).