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Sagot :
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.
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.
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