Answer :
To understand how to graph the piecewise function [tex]\( f(x) = \left\{\begin{array}{c}4 \text { if } x<3 \\ 2x \text { if } x \geq 3\end{array}\right. \)[/tex], let's go through it step-by-step.
### Step 1: Analyze and Graph [tex]\( f(x) \)[/tex] for [tex]\( x < 3 \)[/tex]
For [tex]\( x < 3 \)[/tex], the function is constant:
[tex]\[ f(x) = 4 \][/tex]
This means that for any value of [tex]\( x \)[/tex] that is less than 3, the function takes the value of 4. To graph this segment of the function:
1. Draw a horizontal line at [tex]\( y = 4 \)[/tex] starting from [tex]\( x = -\infty \)[/tex] until [tex]\( x \)[/tex] approaches 3 (not including 3).
### Step 2: Analyze and Graph [tex]\( f(x) \)[/tex] for [tex]\( x \geq 3 \)[/tex]
For [tex]\( x \geq 3 \)[/tex], the function is:
[tex]\[ f(x) = 2x \][/tex]
To graph this segment of the function:
1. Identify the point where [tex]\( x = 3 \)[/tex]:
[tex]\[ f(3) = 2 \cdot 3 = 6 \][/tex]
2. Plot the point (3, 6).
3. Draw a line starting from [tex]\( (3, 6) \)[/tex] extending to the right for all values [tex]\( x \geq 3 \)[/tex].
### Step 3: Determine the Transition Between Segments
- At [tex]\( x = 3 \)[/tex], there is a transition between the two pieces of the function.
- For [tex]\( x < 3 \)[/tex], as mentioned, [tex]\( f(x) = 4 \)[/tex]. We approach [tex]\( x = 3 \)[/tex] from the left side, and [tex]\( f(x) \)[/tex] is 4 for values slightly less than 3.
- For [tex]\( x \geq 3 \)[/tex], [tex]\( f(x) = 2x \)[/tex]. At [tex]\( x = 3 \)[/tex], it reaches 6, creating a jump from 4 to 6 right at [tex]\( x = 3 \)[/tex].
### Step 4: Verify the Transition and Create an Open Circle
Since [tex]\( f(x) \)[/tex] is not continuous at [tex]\( x = 3 \)[/tex]:
- At [tex]\( x = 3 \)[/tex], for the [tex]\( f(x) = 4 \)[/tex] part, we use an open circle (not including the point) at [tex]\( (3, 4) \)[/tex].
- For [tex]\( f(x) = 2x \)[/tex] starting at [tex]\( x = 3 \)[/tex] and [tex]\( f(3) = 6 \)[/tex], we place a solid dot at [tex]\( (3, 6) \)[/tex].
### Step 5: Graph Description Matching
From this analysis, the graph should include:
- A horizontal line at [tex]\( y = 4 \)[/tex] for [tex]\( x < 3 \)[/tex], ending with an open circle at [tex]\( x = 3 \)[/tex].
- A solid dot at [tex]\( x = 3 \)[/tex] and [tex]\( y = 6 \)[/tex].
- An upward sloping line from [tex]\( (3, 6) \)[/tex] relative to the linear equation [tex]\( y = 2x \)[/tex].
Given these points, match the described features with the correct provided text description to select the matching answer choice, considering:
- Open circle at [tex]\( (3, 4) \)[/tex]
- Solid dot at [tex]\( (3, 6) \)[/tex]
- Horizontal line for [tex]\( x < 3 \)[/tex]
- Linear increase for [tex]\( x \geq 3 \)[/tex] with a slope of 2
Then choose the correct corresponding graph description (A, B, or C) that matches these details.
### Step 1: Analyze and Graph [tex]\( f(x) \)[/tex] for [tex]\( x < 3 \)[/tex]
For [tex]\( x < 3 \)[/tex], the function is constant:
[tex]\[ f(x) = 4 \][/tex]
This means that for any value of [tex]\( x \)[/tex] that is less than 3, the function takes the value of 4. To graph this segment of the function:
1. Draw a horizontal line at [tex]\( y = 4 \)[/tex] starting from [tex]\( x = -\infty \)[/tex] until [tex]\( x \)[/tex] approaches 3 (not including 3).
### Step 2: Analyze and Graph [tex]\( f(x) \)[/tex] for [tex]\( x \geq 3 \)[/tex]
For [tex]\( x \geq 3 \)[/tex], the function is:
[tex]\[ f(x) = 2x \][/tex]
To graph this segment of the function:
1. Identify the point where [tex]\( x = 3 \)[/tex]:
[tex]\[ f(3) = 2 \cdot 3 = 6 \][/tex]
2. Plot the point (3, 6).
3. Draw a line starting from [tex]\( (3, 6) \)[/tex] extending to the right for all values [tex]\( x \geq 3 \)[/tex].
### Step 3: Determine the Transition Between Segments
- At [tex]\( x = 3 \)[/tex], there is a transition between the two pieces of the function.
- For [tex]\( x < 3 \)[/tex], as mentioned, [tex]\( f(x) = 4 \)[/tex]. We approach [tex]\( x = 3 \)[/tex] from the left side, and [tex]\( f(x) \)[/tex] is 4 for values slightly less than 3.
- For [tex]\( x \geq 3 \)[/tex], [tex]\( f(x) = 2x \)[/tex]. At [tex]\( x = 3 \)[/tex], it reaches 6, creating a jump from 4 to 6 right at [tex]\( x = 3 \)[/tex].
### Step 4: Verify the Transition and Create an Open Circle
Since [tex]\( f(x) \)[/tex] is not continuous at [tex]\( x = 3 \)[/tex]:
- At [tex]\( x = 3 \)[/tex], for the [tex]\( f(x) = 4 \)[/tex] part, we use an open circle (not including the point) at [tex]\( (3, 4) \)[/tex].
- For [tex]\( f(x) = 2x \)[/tex] starting at [tex]\( x = 3 \)[/tex] and [tex]\( f(3) = 6 \)[/tex], we place a solid dot at [tex]\( (3, 6) \)[/tex].
### Step 5: Graph Description Matching
From this analysis, the graph should include:
- A horizontal line at [tex]\( y = 4 \)[/tex] for [tex]\( x < 3 \)[/tex], ending with an open circle at [tex]\( x = 3 \)[/tex].
- A solid dot at [tex]\( x = 3 \)[/tex] and [tex]\( y = 6 \)[/tex].
- An upward sloping line from [tex]\( (3, 6) \)[/tex] relative to the linear equation [tex]\( y = 2x \)[/tex].
Given these points, match the described features with the correct provided text description to select the matching answer choice, considering:
- Open circle at [tex]\( (3, 4) \)[/tex]
- Solid dot at [tex]\( (3, 6) \)[/tex]
- Horizontal line for [tex]\( x < 3 \)[/tex]
- Linear increase for [tex]\( x \geq 3 \)[/tex] with a slope of 2
Then choose the correct corresponding graph description (A, B, or C) that matches these details.
