Learn it
The domain is every allowed input; the range is every output. Watch for division by zero and square roots of negatives.
Composition: (f ∘ g)(x) = f(g(x)). Work from the inside out.
Worked examples
Example 1
Find the domain of f(x) = √(x − 3).
- Need x − 3 ≥ 0
Answer: x ≥ 3
Example 2
f(x) = 2x + 1 and g(x) = x². Find f(g(3)) and g(f(3)).
- g(3) = 9, f(9) = 19
- f(3) = 7, g(7) = 49
Answer: f(g(3)) = 19; g(f(3)) = 49
Practice
Try these on paper first, then check your work.
- Domain of 1/(x + 2)
- Range of y = x²
- f(x) = x − 4, g(x) = 3x: f(g(x))
- g(f(x)) for the same pair
Show the answer key
- x ≠ −2
- y ≥ 0
- 3x − 4
- 3x − 12