Composite Functions

Mathematics · Algebra · ages 15-16

Name ______________________ Date ____________

The rules are f(x) = 4x and g(x) = x + 2. What is fg(0)?
- 8
- 4
- 2
- 0
The rules are f(x) = x + 2 and g(x) = 3x. What is gf(1)?
- 5
- 7
- 11
- 9
The rules are f(x) = 2x and g(x) = x + 6. What is fg(1)?
- 8
- 14
- 4
- 16
The rules are f(x) = x + 4 and g(x) = 2x. What is fg(1)?
- 6
- 4
- 8
- 10
The rules are f(x) = x squared and g(x) = x + 3. What is gf(2)?
Answer: ______________
The rules are f(x) = x squared + 1 and g(x) = 2x. What is fg(x) as an expression?
- 4x squared + 1
- 2x squared + 1
- 2x squared + 2
- 4x squared + 2
The rules are f(x) = x + 1 and g(x) = 2x + 1. What are fg(2) and gf(2)?
- 6 and 7
- 6 and 6
- 7 and 7
- 7 and 6
The rules are f(x) = 3x and g(x) = x + 1. The composite fg(x) simplifies to 3x + a. What is a?
Answer: ______________
The rules are f(x) = x squared and g(x) = x - 2. What is fg(4) - gf(4)?
Answer: ______________
The rules are f(x) = 4x - 2 and g(x) = x squared - 2. What is fg(2) + gf(2)?
Answer: ______________

Answer key

For grown-ups. Fold this page away before handing over the rest.

8 · g acts first: g(0) is 2. Feed that into f: f(2) is 4 times 2 = 8.
9 · In gf, f acts first: f(1) is 3. Feed that into g: g(3) is 3 times 3 = 9.
14 · g acts first: g(1) is 7. Feed that into f: f(7) is 2 times 7 = 14.
6 · In fg, g acts first: g(1) is 2. Feed that into f: f(2) is 2 + 4 = 6.
7 · In gf, f acts first: f(2) is 4. Feed that into g: g(4) is 4 + 3 = 7.
4x squared + 1 · g acts first: fg(x) = f(2x) = (2x) squared + 1 = 4x squared + 1. Squaring 2x squares the 2 as well.
6 and 7 · fg(2): g(2) is 5, f(5) is 6. gf(2): f(2) is 3, g(3) is 7. So fg(2) is 6 and gf(2) is 7.
3 · g acts first: fg(x) = f(x + 1) = 3(x + 1) = 3x + 3, so a is 3.
-10 · fg(4): g(4) is 2, f(2) is 4. gf(4): f(4) is 16, g(16) is 14. The difference is 4 - 14 = -10.
40 · fg(2): g(2) is 2, f(2) is 6. gf(2): f(2) is 6, g(6) is 34. The sum is 6 + 34 = 40.