Find the information you're looking for at Westonci.ca, the trusted Q&A platform with a community of knowledgeable experts. Get quick and reliable answers to your questions from a dedicated community of professionals on our platform. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
Sure, let's carefully analyze, plot, and examine the function [tex]\( f(x) \)[/tex].
The function [tex]\( f(x) \)[/tex] is defined in two different pieces:
[tex]\[ f(x) = \begin{cases} -x + 3 & \text{if } x < -1 \\ -3x + 1 & \text{if } x \geq -1 \end{cases} \][/tex]
### Step 1: Plotting the Function
1. For [tex]\( x < -1 \)[/tex]:
- The function is [tex]\( f(x) = -x + 3 \)[/tex].
- This is a linear function with a slope of -1 and a y-intercept of 3.
2. For [tex]\( x \geq -1 \)[/tex]:
- The function is [tex]\( f(x) = -3x + 1 \)[/tex].
- This is also a linear function, but with a slope of -3 and a y-intercept of 1.
### Step 2: Examine the Behavior at the Point [tex]\( x = -1 \)[/tex]
#### Left-Hand Side for [tex]\( x < -1 \)[/tex]:
[tex]\[ \lim_{{x \to -1^-}} f(x) = -(-1) + 3 = 1 + 3 = 4 \][/tex]
#### Right-Hand Side for [tex]\( x \geq -1 \)[/tex]:
[tex]\[ \lim_{{x \to -1^+}} f(x) = -3(-1) + 1 = 3 + 1 = 4 \][/tex]
Both the left-hand limit and the right-hand limit as [tex]\( x \)[/tex] approaches [tex]\(-1\)[/tex] are equal to 4. Thus,
[tex]\[ \lim_{{x \to -1^-}} f(x) = \lim_{{x \to -1^+}} f(x) = 4 \][/tex]
Also, we need to ensure the function value at [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = -3(-1) + 1 = 3 + 1 = 4 \][/tex]
Since [tex]\( \lim_{{x \to -1^-}} f(x) = \lim_{{x \to -1^+}} f(x) = f(-1) = 4 \)[/tex], the function is indeed continuous at [tex]\( x = -1 \)[/tex].
### Step 3: Graphing the Function
To graph the function properly, we consider the two intervals:
1. For [tex]\( x < -1 \)[/tex] (line segment for [tex]\( f(x) = -x + 3 \)[/tex]):
- It passes through the point [tex]\((-1, 4)\)[/tex] but does not include [tex]\((-1, 4)\)[/tex] (open circle).
2. For [tex]\( x \geq -1 \)[/tex] (line segment for [tex]\( f(x) = -3x + 1 \)[/tex]):
- It starts at [tex]\((-1, 4)\)[/tex] including the point (closed circle) and continues.
Below is a descriptive plot:
- An open circle at [tex]\((-1, 4)\)[/tex] for the first piece ([tex]\(x < -1\)[/tex]).
- A filled circle at [tex]\((-1, 4)\)[/tex] for the second piece ([tex]\(x \geq -1\)[/tex]).
### Final Result
- The function [tex]\( f(x) \)[/tex] is continuous for all [tex]\( x \in \mathbb{R} \)[/tex].
The correct graph of [tex]\( f(x) \)[/tex] includes:
- A line with slope -1 starting from [tex]\((-\infty, 3)\)[/tex], passing [tex]\((-1, 4)\)[/tex] but not including it (open circle).
- Another line with slope -3 starting at [tex]\((-1, 4)\)[/tex] including it (closed circle), and extending to [tex]\( (+\infty, -\infty) \)[/tex].
So, we have successfully graphed the function and confirmed its continuity at [tex]\( x = -1 \)[/tex].
The function [tex]\( f(x) \)[/tex] is defined in two different pieces:
[tex]\[ f(x) = \begin{cases} -x + 3 & \text{if } x < -1 \\ -3x + 1 & \text{if } x \geq -1 \end{cases} \][/tex]
### Step 1: Plotting the Function
1. For [tex]\( x < -1 \)[/tex]:
- The function is [tex]\( f(x) = -x + 3 \)[/tex].
- This is a linear function with a slope of -1 and a y-intercept of 3.
2. For [tex]\( x \geq -1 \)[/tex]:
- The function is [tex]\( f(x) = -3x + 1 \)[/tex].
- This is also a linear function, but with a slope of -3 and a y-intercept of 1.
### Step 2: Examine the Behavior at the Point [tex]\( x = -1 \)[/tex]
#### Left-Hand Side for [tex]\( x < -1 \)[/tex]:
[tex]\[ \lim_{{x \to -1^-}} f(x) = -(-1) + 3 = 1 + 3 = 4 \][/tex]
#### Right-Hand Side for [tex]\( x \geq -1 \)[/tex]:
[tex]\[ \lim_{{x \to -1^+}} f(x) = -3(-1) + 1 = 3 + 1 = 4 \][/tex]
Both the left-hand limit and the right-hand limit as [tex]\( x \)[/tex] approaches [tex]\(-1\)[/tex] are equal to 4. Thus,
[tex]\[ \lim_{{x \to -1^-}} f(x) = \lim_{{x \to -1^+}} f(x) = 4 \][/tex]
Also, we need to ensure the function value at [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = -3(-1) + 1 = 3 + 1 = 4 \][/tex]
Since [tex]\( \lim_{{x \to -1^-}} f(x) = \lim_{{x \to -1^+}} f(x) = f(-1) = 4 \)[/tex], the function is indeed continuous at [tex]\( x = -1 \)[/tex].
### Step 3: Graphing the Function
To graph the function properly, we consider the two intervals:
1. For [tex]\( x < -1 \)[/tex] (line segment for [tex]\( f(x) = -x + 3 \)[/tex]):
- It passes through the point [tex]\((-1, 4)\)[/tex] but does not include [tex]\((-1, 4)\)[/tex] (open circle).
2. For [tex]\( x \geq -1 \)[/tex] (line segment for [tex]\( f(x) = -3x + 1 \)[/tex]):
- It starts at [tex]\((-1, 4)\)[/tex] including the point (closed circle) and continues.
Below is a descriptive plot:
- An open circle at [tex]\((-1, 4)\)[/tex] for the first piece ([tex]\(x < -1\)[/tex]).
- A filled circle at [tex]\((-1, 4)\)[/tex] for the second piece ([tex]\(x \geq -1\)[/tex]).
### Final Result
- The function [tex]\( f(x) \)[/tex] is continuous for all [tex]\( x \in \mathbb{R} \)[/tex].
The correct graph of [tex]\( f(x) \)[/tex] includes:
- A line with slope -1 starting from [tex]\((-\infty, 3)\)[/tex], passing [tex]\((-1, 4)\)[/tex] but not including it (open circle).
- Another line with slope -3 starting at [tex]\((-1, 4)\)[/tex] including it (closed circle), and extending to [tex]\( (+\infty, -\infty) \)[/tex].
So, we have successfully graphed the function and confirmed its continuity at [tex]\( x = -1 \)[/tex].
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.
