Given the function complete the following

Question

 f(x)={x+3   if x<-2, |x-1|-3  if -2≤x<3,   -1  if x>3

 

a. Graph the function.

 

Based on the graph, find the following values:

 

b. lim_x→-2^- f(x)

c. lim_x→-2⁺ f(x)

d. lim_x→-2 f(x)

e. lim_x→3- f(x)

f. lim_x→3⁺ f(x)

g. lim_x→3 f(x)

h. lim_x→2- f(x)

i. lim_x→2⁺ f(x)

j. lim_x→2 f(x)

Answer 1

Go ahead and graph this piece-wise function.

 

The first piece will be the straight line y = x + 3 and will go from -infinity to -2. Pick a few points to see what it looks like ((-4, -1), (-3, 0)), and continue the line up until (-2, 1). Graph an open circle there to indicate that that point isn't included in your function.

 

The second piece will be the v-shaped absolute value function y = |x - 1| - 3, which will go from -2 to 3. Pick a few points to see what it looks like ((-2, 0), (0, -2), (1, -3), (2, -2)) and continue the v-shape up until (3, -1). Graph an open circle there to indicate that that point isn't included in your function.

 

The last piece will be the horizontal line y = -1 and will go from 3 to infinity. Fill in the point (3, -1) since it is included in the function now, and extend the horizontal line out through (4, -1), etc.

 

 

 

The limit questions are just asking you what's happening to the value of f(x) as we approach the given value of x.

 

(b) We're approaching -2 from the negative direction (from the left). As the function approaches -2 from the left, we're on the line y = x + 3. So, the limit is f(x) = -2 + 3 = 1.

 

(c) We're approaching -2 from the positive direction (from the right). As the function approaches -2 from the right, we're on the line y = |x - 1| - 3. So, the limit is f(x) = |-2 - 1| - 3 = |-3| - 3 = 3 - 3 = 0.

 

(d) We're approaching -2 with no direction given. Since the positive and negative limits are not equal (1 and 0), this limit is undefined.

 

Continue doing this for the rest of the limits.