Let the function f be defined by:
f(x)={x+66if x<−3if x>−3.
Sketch the graph of this function and find the following limits, if they exist
first, finish defining f(x), and oh yeah, you might actually include a question...
To sketch the graph of the function f(x), we can start by plotting some key points.
1. When x < -3:
In this case, according to the function definition, f(x) = x + 6. So, let's plot some points for x values less than -3. For example, for x = -4:
f(-4) = -4 + 6 = 2. So, we have the point (-4, 2).
2. When x > -3:
In this case, according to the function definition, f(x) = -3. So, let's plot some points for x values greater than -3. For example, for x = 0:
f(0) = -3. So, we have the point (0, -3).
Now, let's plot these points on a graph:
^
|
|
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| *
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| -3
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-4 0
Next, we can notice that the graph consists of a line passing through the point (-4, 2) with a slope of 1 until we reach x = -3. At x = -3, the graph has a vertical jump and remains constant with a height of -3.
Now, let's find the limits, if they exist:
1. Limit as x approaches -3 from the left (x → -3-):
This means we approach -3 from values that are less than -3. From the graph, we can see that as we approach -3 from the left, the function approaches 2. Hence, the limit is 2.
2. Limit as x approaches -3 from the right (x → -3+):
This means we approach -3 from values that are greater than -3. From the graph, we can see that as we approach -3 from the right, the function remains constant at -3. Hence, the limit is -3.
In summary, the limits are:
- Limit as x approaches -3 from the left (x → -3-): 2
- Limit as x approaches -3 from the right (x → -3+): -3