Homework:

Functions Worksheet 4, Functions Assignment 2

Tips (warning major spoilers ahead):

Worksheet 4
Extension 2, 3: Let f(x) = y. Therefore gf(x) = g(y) = (x^2 + 6x + 2), for the first question. and what does y equal? f(x), which is (x + 3) for the first question. Sub in and solve.

Extension 6: Same as above, but combine g(x) and f(x) first.

Extension 4,5: Slightly different. Let k(x) = y. Therefore h(y) = x^2 + 6x + 11 = y^2 + 2. The middle expression is from the question, since h(k(x)) = h(y). The last bit is if you just subbed in y into h(x) as if y was a constant. Solve.

Assignment 2
Q5: Don't forget that a > 0.
Q6, 7: Same as Worksheet 4.

Answers (For checking):

Worksheet 4
Example 3: (fg)^(-3) = 37
Example 4: (a) k = -2, (b) fg(x) = 3/(2x + 3), x not equals (-3/2), (c) p = 1/7
Example 5: a = 9

1(a) f^(-1)(x) = (-3x - 1)/(x - 3)
(b) x = 11/3
(c) f^(-1)g^(-1)(g) = -29/3, (fg)^(-1)(7) = -31/12
2 g(x) = x^2 - 7
3 k(x) = x^2 - 3
4 k(x) = plus/minus (x + 3)
5 g(x) = (6x - 1)/(2 - 2x), x not equals 1
6 h(x) = y - 3

Assignment 2
1 (a) x = -3, (b) x = -3
2 (a) fg(x) = (x - 1)/(x + 1), x not equals -1; gf(x) = (2x - 1)/2x, x not equals 0; (b) x = 1/3
3 k = 2 or k = 1
4 f^n(x) = x/(nx + 1)
5 a = 1, b = 2
6 h(x) = 3x^2 - 2x - 8
7 (a) g(x) = x^2 + x + 2
(b) g(x) = (4x^2 + 3x - 18)/2