Inequalities and other gr.12 calculus review

[AzureNight]

Argh, I haven't done any calculus since gr. 12 and university calculus is killing me. Need some help with a few questions; and I know there are a few that I'm asking for on here, but they are only a few out of many (I've done most).

Homework Statement

1. Solve the following inequality. (graph on a number line - I'll do that myself)

|x - 3| < |2x + 1| - would I have to do four "cases"? Surely there is a more efficient way to solve it?

2. Sketch the graph of the following functions:

f(x) = 1 + cos(3x + pi/2), where x is the element of [-pi/2, pi/2] - I guess you guys can't help me with the graph, but some hints on what to do with this function would be appreciated. I have not worked with such a function in grade 12 (for graphing), so I have no clue where to begin.

3. determine whether the following statements are true or false for all functions f,g, and h. Justify your answer with an appropriate proof or counter example.

i) (f + g) o h = f o h + g o h - read "f o h" as "f(h(x))".

4. Define the functions:

fE(x) = f(x) + f(-x), fO(x) = f(x) - f(-x)

a) show that fE(x) is even and fO(x) is odd (what? I don't even understand what they mean by that)

b) by using the result in part a), prove that any function can be written as the sum of two functions, one of which is even and the other odd.

The Attempt at a Solution

Unfortunately these questions have me stumped; I don't know where to start. :(

For 3. i) however, I showed that the statement cannot be disproven by a counter example, by substituting 3 arbitrary functions. There was a part ii) statement, which I immediately disproved with a counter example.

 

[Avodyne]

|x - 3| < |2x + 1| - would I have to do four "cases"?

Yes

Sketch the graph of the following functions: f(x) = 1 + cos(3x + pi/2)

Can you graph cos(x)? If so, can you graph 1+cos(x)? Can you graph cos(3x)? Can you graph cos(x+a), where "a" is a positive contstant? Figure these out, then figure out how to put them together.

(f + g) o h = f o h + g o h. I showed that the statement cannot be disproven by a counter example, by substituting 3 arbitrary equations.

I don't know what you mean; can you show your work?

fE(x) = f(x) + f(-x), fO(x) = f(x) - f(-x)
a) show that fE(x) is even and fO(x) is odd (what? I don't even understand what they mean by that)

A function f(x) is even if f(-x)=f(x), and odd if f(-x)=-f(x).

b) by using the result in part a), prove that any function can be written as the sum of two functions, one of which is even and the other odd.

Can you express f(x) in terms of fE(x) and fO(x)?

 

[HallsofIvy]

3. determine whether the following statements are true or false for all functions f,g, and h. Justify your answer with an appropriate proof or counter example.

i) (f + g) o h = f o h + g o h - read "f o h" as "f(h(x))".

For 3. i) however, I showed that the statement cannot be disproven by a counter example, by substituting 3 arbitrary functions. There was a part ii) statement, which I immediately disproved with a counter example.

How did you prove their couldn't be counter example? Just showing it is true for 3 arbitrarily choosen functions doesn't mean that some other three functions won't show it isn't true. The only way to show a statement "cannot be disproven by a counter example" is to prove it! What is the definition of f+ g?