- 3

- 0

**1. Homework Statement**

f(x) = 4 for x > or = 0, f(x) = 0 for x < 0, and g(x) = x^2 for all x.

Thus dom(f) = dom(g) = R.

**2. Homework Equations**

a. Determine the following functions: f+g, fg, f o g, g o f. Be sure to specify thier domain.

b. Which of the functions f, g, f+g, fg, f o g, g o f is continuous

**3. The Attempt at a Solution**

Ok, so for part (a) I am at f o g step, which I say is f(x^2) = 4 for x > or = 0 and f(x^2) = 0 for x < 0.

Question 1. Since f is a function of g(x), then when I restrict it values, do I have to restrict them in terms of x (as given) or in terms of x^2, since it is a f of g(x) now? Would I have to say that f(x^2) = 4 for x^2 > or = 0 and f(x^2) = 0 for x^2 < 0, instead of how I provided above? If I do have to say it in terms of x^2, then x^2 < 0 makes no sense, and the f would not be defined as f(g(x)) < 0?

Question 2. Since the composition f o g from Q1 is not piecewise anymore and starts at 0 and goes to positive infinity, would it make the composition continuous function?

Question 3. I am a little confused on how to prove that f is not a continuous function using limits. Would someone be able to show me a quick example please?

Question 4. If f is not continuous, then I assume that f, g, f+g, fg are not continuous either. But if f o g IS continuous (Q3), then wouldnt it contradict a Theorem that states "If f is continuous at x0 and g is continuous at f(x0), then the composite functions g o f is continuous at x0"?

Thank you for your time.