Question

Sketch a function with the following characteristics?

• g (x) is defined at x = -2
• g (x) is continuous for all real numbers except at x = -2 and x = 1
• (Limit x -> -2) g(x) = 3
• g'(3) DNE
• g(0) = -4
• g'(x) = 0 for x > 3

Answer 1

One such graph would be the following:

Firs of all we're given that `g(x)` is defined at `x = -2`, that `g(x)` is not continuous at `x = -2` and `lim_(x-> -2) g(x) = 3`. As a result, we put a blank point at `(-2, 2.5)` and we put a point at `(-2, 3)`.

We then need to ensure that `(0, -4)` lies on the graph. Following this, we make `g(x)` discontinuous by putting a hole at `x = 1`. For `g'(3)` to not exist, we probably need a cusp. As required, I put it on. The graph is not differentiable there because there is more than one possible tangent (it was supposed to be like a 90˚ angle).

The final condition is for `g'(x) = 0` for `x >3`, so the tangent will be horizontal, as shown above.

Hopefully this helps!