Homework 1

Homework 1, 9.2: #1, 2, 3; bonus: #4

75 pts + 25 points bonus

(1) 25 pts: see Nancy's solution

sgn(x)f(x) means multiplication, not composition
students were good at noting that sgn(x) is not defined at x=0
In part (a) many students knew (somehow...) that when x<0, the graph f(x) was reflected across the x-axis, but most didn't give sufficient explanation.
In part (b), most students assumed that f(x) was negative when x was, and f(x) was positive when x was. I don't know where this assumption came from, but you should solve the problem in general for any f(x).
In both parts, pictures are very helpful in understanding the problem youself, as well as explaining it to me.

(2) 25 pts: see Tomoko's solution

I wanted to see an explicit counter-example.
Pictures are fine, but I wanted to see f, g, and fg. Many students showed a single graph of something, without explanation, and called it a counter-example.

(3) 25 pts: see Nancy's solution

(a) was done very well by all students
(b) was mostly good, but students neglected to give the domain they were equal on.
(c) most students noted that sqrt(x)=f(x) is not defined for x<0, but only a few students noted that sqrt(x^2)=g o f (x) is actually equal to |x|, not x.

bonus (4) 25 pts: see part (c) of Tomoko's solution for an excellent example.