To graph the given piecewise function, we'll consider each piece of the function separately and graph them according to their defined domains. Here is a step-by-step approach:
1. Identify the pieces of the function and their domains:
- [tex]\( f(x) = 3x - 5 \)[/tex] for [tex]\( x \leq -1 \)[/tex]
- [tex]\( f(x) = -2x + 3 \)[/tex] for [tex]\( -1 < x < 4 \)[/tex]
- [tex]\( f(x) = 2 \)[/tex] for [tex]\( x \geq 4 \)[/tex]
2. Graph each piece within its domain:
- For [tex]\( f(x) = 3x - 5 \)[/tex] [tex]\( (x \leq -1) \)[/tex]:
[tex]\[
\begin{aligned}
&\text{We consider the range of } x \text{ values from } -10 \text{ to } -1.\\
&\text{Calculate } y \text{ values for } each \text{ } x \text{ } value.\\
& f(-10) = 3(-10) - 5 = -35 \\
& f(-9.9) = 3(-9.9) - 5 = -34.7 \\
& f(-9.8) = 3(-9.8) - 5 = -34.4 \\
& \vdots \\
& f(-1) = 3(-1) - 5 = -8
\end{aligned}
\][/tex]
The points will be [tex]\((x, y) = (-10, -35), (-9.9, -34.7), \ldots, (-1, -8)\)[/tex].
- For [tex]\( f(x) = -2x + 3 \)[/tex] [tex]\( (-1 < x < 4) \)[/tex]:
[tex]\[
\begin{aligned}
&\text{We consider the range of } x \text{ values from } -0.9 \text{ to } \text{ just below } 4.\\
&\text{Calculate } y \text{ values for } each \text{ x value.}\\
& f(-0.9) = -2(-0.9) + 3 = 4.8 \\
& f(-0.8) = -2(-0.8) + 3 = 4.6 \\
& f(-0.7) = -2(-0.7) + 3 = 4.4 \\
& \vdots \\
& f(3.9) = -2(3.9) + 3 = -4.8
\end{aligned}
\][/tex]
The points will be [tex]\((x, y) = (-0.9, 4.8), (-0.8, 4.6), \ldots, (3.9, -4.8)\)[/tex].
- For [tex]\( f(x) = 2 \)[/tex] [tex]\( (x \geq 4) \)[/tex]:
[tex]\[
\begin{aligned}
&\text{We consider the range of } x \text{ values starting just above } 4 \text{ to 10.}\\
& f(4) = 2 \\
& f(4.1) = 2 \\
& f(4.2) = 2 \\
& \vdots \\
& f(9.9) = 2
\end{aligned}
\][/tex]
The points will be [tex]\((x, y) = (4, 2), (4.1, 2), \ldots, (9.9, 2)\)[/tex].
3. Plot the points and draw the graph:
- Plot the points [tex]\((x, y) = (-10, -35), (-9.9, -34.7), \ldots, (-1, -8)\)[/tex] and draw a line through them for [tex]\( f(x) = 3x - 5 \)[/tex] in the domain [tex]\( x \leq -1 \)[/tex].
- Plot the points [tex]\((x, y) = (-0.9, 4.8), (-0.8, 4.6), \ldots, (3.9, -4.8)\)[/tex] and draw a line through them for [tex]\( f(x) = -2x + 3 \)[/tex] in the domain [tex]\( -1 < x < 4 \)[/tex].
- Plot the points [tex]\((x, y) = (4, 2), (4.1, 2), \ldots, (9.9, 2)\)[/tex] and draw a constant line for [tex]\( f(x) = 2 \)[/tex] in the domain [tex]\( x \geq 4 \)[/tex].
This process will give us the complete graph of the piecewise function, clearly showing each segment defined by the given conditions.
