Answer :
Certainly! Let's break down the problem step-by-step and graph the piecewise function [tex]\( f(x) \)[/tex].
### 1. Understanding the Piecewise Function
The function [tex]\( f(x) \)[/tex] is defined differently over three intervals:
1. For [tex]\( x \leq -1 \)[/tex], the function is: [tex]\( f(x) = 3x - 5 \)[/tex].
2. For [tex]\( -1 < x < 4 \)[/tex], the function is: [tex]\( f(x) = -2x + 3 \)[/tex].
3. For [tex]\( x \geq 4 \)[/tex], the function is: [tex]\( f(x) = 2 \)[/tex].
### 2. Determine Key Points and Behavior
First Interval: [tex]\( x \leq -1 \)[/tex]
For [tex]\( f(x) = 3x - 5 \)[/tex]:
- When [tex]\( x = -1 \)[/tex], [tex]\( f(-1) = 3(-1) - 5 = -3 - 5 = -8 \)[/tex].
- As [tex]\( x \)[/tex] becomes more negative (decreasing further), the value of [tex]\( f(x) \)[/tex] becomes more negative.
Second Interval: [tex]\( -1 < x < 4 \)[/tex]
For [tex]\( f(x) = -2x + 3 \)[/tex]:
- When [tex]\( x = -1 \)[/tex], technically, this doesn't apply, but approaching from the right, consider [tex]\( x \)[/tex] values slightly greater than [tex]\(-1\)[/tex].
- When [tex]\( x = 0 \)[/tex], [tex]\( f(0) = -2(0) + 3 = 3 \)[/tex].
- When [tex]\( x = 4 \)[/tex], [tex]\( f(4) = -2(4) + 3 = -8 + 3 = -5 \)[/tex] (approaching but not reaching this value exactly, as [tex]\( x \)[/tex] is strictly less than 4).
Third Interval: [tex]\( x \geq 4 \)[/tex]
For [tex]\( f(x) = 2 \)[/tex]:
- At exactly [tex]\( x = 4 \)[/tex], [tex]\( f(x) \)[/tex] shifts to a constant value.
- For [tex]\( x \)[/tex] values greater than 4, [tex]\( f(x) \)[/tex] remains at 2.
### 3. Plot the Points and Graph Segments
To graph the piecewise function:
1. First Interval ([tex]\( x \leq -1 \)[/tex]):
- Plot several points such as [tex]\( (-1, -8) \)[/tex], [tex]\( (-2, -11) \)[/tex], etc.
- Draw a straight line through these points.
2. Second Interval ([tex]\( -1 < x < 4 \)[/tex]):
- Draw a straight line segment between calculated points.
- Example points: [tex]\( (-0.5, 4) \)[/tex], [tex]\( (0, 3) \)[/tex], [tex]\( (1, 1) \)[/tex], [tex]\( (2, -1) \)[/tex], and [tex]\( (3, -3) \)[/tex].
3. Third Interval ([tex]\( x \geq 4 \)[/tex]):
- The function is always equal to 2 for [tex]\( x \geq 4 \)[/tex].
- Plot a horizontal line starting from [tex]\( (4, 2) \)[/tex] and extending to the right.
### Graph
To put these into a visual form:
1. For [tex]\( x \leq -1 \)[/tex]: Plot the line [tex]\( y = 3x - 5 \)[/tex] up to and including the point [tex]\( (-1, -8) \)[/tex].
2. For [tex]\( -1 < x < 4 \)[/tex]: Plot the line segment [tex]\( y = -2x + 3 \)[/tex] from just to the right of [tex]\(-1\)[/tex] up to just before [tex]\( x = 4 \)[/tex].
3. For [tex]\( x \geq 4 \)[/tex]: Draw a horizontal line at [tex]\( y = 2 \)[/tex] starting from the point [tex]\( (4, 2) \)[/tex].
### Graph Illustration:
1. First Interval:
- Line Segment from [tex]\((-\infty, \infty) \to (-1, -8)\)[/tex]
2. Second Interval:
- Line from just above [tex]\(( -1, \text{ not defined} ) \to (4, -5) \)[/tex]
3. Third Interval:
- Line Segment at [tex]\((4, 2) \to (\infty, 2)\)[/tex]
[tex]\[ \begin{array}{c} \begin{tikzpicture}[domain=-5:5] \draw[very thin,color=gray] (-5.1,-9.1) grid (5.1,3.1); \draw[->] (-5.1,0) -- (5.2,0) node[right] {$x$}; \draw[->] (0,-9.1) -- (0,3.2) node[above] {$f(x)$}; % Plot the first part of the piecewise function \draw[color=red, line width=0.9pt] (-5,-20) -- (-1,-8); % Plot the second part of the piecewise function \draw[color=blue, line width=0.9pt, domain=-1:4] plot (\x, {(-2)*\x + 3}); % Plot the third part of the piecewise function \draw[color=green, line width=0.9pt] (4,2) -- (5,2); % Label points of importance \fill[black] (-1,-8) circle (2pt) node[anchor=north] {(-1, -8)}; \fill[black] (4,2) circle (2pt) node[anchor=west] {(4, 2)}; \end{tikzpicture} \end{array} \][/tex]
This visual should help you see how the piecewise function behaves on different intervals.
### 1. Understanding the Piecewise Function
The function [tex]\( f(x) \)[/tex] is defined differently over three intervals:
1. For [tex]\( x \leq -1 \)[/tex], the function is: [tex]\( f(x) = 3x - 5 \)[/tex].
2. For [tex]\( -1 < x < 4 \)[/tex], the function is: [tex]\( f(x) = -2x + 3 \)[/tex].
3. For [tex]\( x \geq 4 \)[/tex], the function is: [tex]\( f(x) = 2 \)[/tex].
### 2. Determine Key Points and Behavior
First Interval: [tex]\( x \leq -1 \)[/tex]
For [tex]\( f(x) = 3x - 5 \)[/tex]:
- When [tex]\( x = -1 \)[/tex], [tex]\( f(-1) = 3(-1) - 5 = -3 - 5 = -8 \)[/tex].
- As [tex]\( x \)[/tex] becomes more negative (decreasing further), the value of [tex]\( f(x) \)[/tex] becomes more negative.
Second Interval: [tex]\( -1 < x < 4 \)[/tex]
For [tex]\( f(x) = -2x + 3 \)[/tex]:
- When [tex]\( x = -1 \)[/tex], technically, this doesn't apply, but approaching from the right, consider [tex]\( x \)[/tex] values slightly greater than [tex]\(-1\)[/tex].
- When [tex]\( x = 0 \)[/tex], [tex]\( f(0) = -2(0) + 3 = 3 \)[/tex].
- When [tex]\( x = 4 \)[/tex], [tex]\( f(4) = -2(4) + 3 = -8 + 3 = -5 \)[/tex] (approaching but not reaching this value exactly, as [tex]\( x \)[/tex] is strictly less than 4).
Third Interval: [tex]\( x \geq 4 \)[/tex]
For [tex]\( f(x) = 2 \)[/tex]:
- At exactly [tex]\( x = 4 \)[/tex], [tex]\( f(x) \)[/tex] shifts to a constant value.
- For [tex]\( x \)[/tex] values greater than 4, [tex]\( f(x) \)[/tex] remains at 2.
### 3. Plot the Points and Graph Segments
To graph the piecewise function:
1. First Interval ([tex]\( x \leq -1 \)[/tex]):
- Plot several points such as [tex]\( (-1, -8) \)[/tex], [tex]\( (-2, -11) \)[/tex], etc.
- Draw a straight line through these points.
2. Second Interval ([tex]\( -1 < x < 4 \)[/tex]):
- Draw a straight line segment between calculated points.
- Example points: [tex]\( (-0.5, 4) \)[/tex], [tex]\( (0, 3) \)[/tex], [tex]\( (1, 1) \)[/tex], [tex]\( (2, -1) \)[/tex], and [tex]\( (3, -3) \)[/tex].
3. Third Interval ([tex]\( x \geq 4 \)[/tex]):
- The function is always equal to 2 for [tex]\( x \geq 4 \)[/tex].
- Plot a horizontal line starting from [tex]\( (4, 2) \)[/tex] and extending to the right.
### Graph
To put these into a visual form:
1. For [tex]\( x \leq -1 \)[/tex]: Plot the line [tex]\( y = 3x - 5 \)[/tex] up to and including the point [tex]\( (-1, -8) \)[/tex].
2. For [tex]\( -1 < x < 4 \)[/tex]: Plot the line segment [tex]\( y = -2x + 3 \)[/tex] from just to the right of [tex]\(-1\)[/tex] up to just before [tex]\( x = 4 \)[/tex].
3. For [tex]\( x \geq 4 \)[/tex]: Draw a horizontal line at [tex]\( y = 2 \)[/tex] starting from the point [tex]\( (4, 2) \)[/tex].
### Graph Illustration:
1. First Interval:
- Line Segment from [tex]\((-\infty, \infty) \to (-1, -8)\)[/tex]
2. Second Interval:
- Line from just above [tex]\(( -1, \text{ not defined} ) \to (4, -5) \)[/tex]
3. Third Interval:
- Line Segment at [tex]\((4, 2) \to (\infty, 2)\)[/tex]
[tex]\[ \begin{array}{c} \begin{tikzpicture}[domain=-5:5] \draw[very thin,color=gray] (-5.1,-9.1) grid (5.1,3.1); \draw[->] (-5.1,0) -- (5.2,0) node[right] {$x$}; \draw[->] (0,-9.1) -- (0,3.2) node[above] {$f(x)$}; % Plot the first part of the piecewise function \draw[color=red, line width=0.9pt] (-5,-20) -- (-1,-8); % Plot the second part of the piecewise function \draw[color=blue, line width=0.9pt, domain=-1:4] plot (\x, {(-2)*\x + 3}); % Plot the third part of the piecewise function \draw[color=green, line width=0.9pt] (4,2) -- (5,2); % Label points of importance \fill[black] (-1,-8) circle (2pt) node[anchor=north] {(-1, -8)}; \fill[black] (4,2) circle (2pt) node[anchor=west] {(4, 2)}; \end{tikzpicture} \end{array} \][/tex]
This visual should help you see how the piecewise function behaves on different intervals.
