Key Terms
EXAMPLE
F(x) = 2/(x - 3) + 4
Domain
All real numbers except x = 1
Read as
"f composed with g at x" or "f of g of x."
DEFINITION (formal)
The domain of (f o g) is all x such that:
Domain of g
X <= 3, so (-inf, 3] Composite: (f o g)(x) = sqrt(sqrt(3 - x) + 2)
Domain of f
[4, inf). Range of f: [2, inf).
EXAMPLE with radicals
F(x) = sqrt(x + 2) and g(x) = sqrt(3 - x)
Decomposition is the reverse of composition
Rewrite a complicated function as f(g(x)) using two simpler functions.
STRATEGY
Look for an inner expression that appears as a chunk inside the formula. That chunk becomes the inner function.
Even function
F(x) = f(-x); symmetric about the y-axis.
Odd function
F(x) = -f(-x); symmetric about the origin.
Neither
Most functions. Example: f(x) = 2^x
VERTICAL
In the form a*f(x) + k Apply stretch/compression (multiply by a) FIRST, then vertical shift (add k).
HORIZONTAL
In the form f(bx + h)
Factor first
F(b(x + h/b)) Apply horizontal stretch (1/b) FIRST, then horizontal shift.