Which of the following definitions of g(x) and A would make f(x) continuous on R?

Question

Consider f(x) be defined by the piecewise function. Which of the following definitions of g(x) and A would make f(x) continuous on R?

f(x) = -2x - 2, x ∈ (- infinity, - 1 )

g(x), x ∈ A

3 , x ∈ [2 , infinity)

a) g(x) = x² - 1, A = [-1,2)

b) g(x) = x² - 1, A = (-1,2]

c) g(x) = x³ + 1, A = [-1,2)

d) g(x) = x³ + 1, A = (-1,2]

Answer 1

A continuous function is well defined at every point, such that the left and right limits are equal. What does this mean? It means if you follow the function toward x = a from the left you will hit the same point you would hit if you followed it to x = from the right. In other words, there's no "jump" at x = a in the value of the function.

Step 1, determine which functions g(x) match the values at the endpoints (x = -1 and x = 2). At the former, we have f(x) = -2x - 2. Compute the value at x = -1. At the latter, we are told the value is 3. Do any of the definitions of g(x) match these values at x = -1 and and x = 2?

Step 2, assuming one or more do (hint: one or more do), we need to determine if the range of the definition of g(x) matches what we are given. There are two ways of specifying a range. One uses "(" or ")". The other uses "[" or "]". What's the difference? If I say that x is in (a, b), I'm saying a < x < b. If I say that x is in [a,b], I'm saying that a <= x <= b. The first is an "open" interval. The latter is "closed."

A piecewise continuous function is defined for all x. Thus we couldn't have something like:

x = 2 for x in (a, b)

x = 2 + (x - b) in (b, c)

because neither range includes x = b. What would work is, for example:

x = 2 in (a, b)

x = 2 + (x - b) in [b, c)

With these concepts, you can answer the question.