Question: Type the correct answer in each box. Use numerals instead of words. If necessary, use / for the fraction bar(s).
Graph shows a piecewise function plotted on a coordinate plane. The function has two segments. One segment has closed endpoints at $(3, 2)$ and $(5, 4)$. The other segment has closed endpoints at $(-1, 4)$ and $(1, 4)$.
The graph represents the piecewise function:
$$f(x) = \begin{cases} \text{Text entry field 1 of 2 Enter answer here}, & \text{if } -1 \leq x \leq 1 \\ \text{Text entry field 2 of 2 Enter answer here}, & \text{if } 3 \leq x \leq 5 \end{cases}$$
1. **State the problem:** We need to find the piecewise function $f(x)$ defined on two intervals: $[-1,1]$ and $[3,5]$ based on the given points.
2. **Analyze the first segment:** The segment from $(-1,4)$ to $(1,4)$ is a horizontal line because the $y$-values are constant at $4$.
Therefore, for $-1 \leq x \leq 1$,
$$f(x) = 4$$
3. **Analyze the second segment:** The segment from $(3,2)$ to $(5,4)$ is a line with endpoints given.
Find the slope $m$:
$$m = \frac{4 - 2}{5 - 3} = \frac{2}{2} = 1$$
Use point-slope form with point $(3,2)$:
$$y - 2 = 1(x - 3)$$
$$y = x - 3 + 2 = x - 1$$
So for $3 \leq x \leq 5$,
$$f(x) = x - 1$$
4. **Write the piecewise function:**
$$f(x) = \begin{cases} 4, & -1 \leq x \leq 1 \\ x - 1, & 3 \leq x \leq 5 \end{cases}$$
**Final answer:**
$$f(x) = \begin{cases} 4, & -1 \leq x \leq 1 \\ x - 1, & 3 \leq x \leq 5 \end{cases}$$