Notation & Core Definitions
- \( (f + g)(x) = f(x) + g(x) \) (always use parentheses for combined functions)
- \( (f - g)(x) = f(x) - g(x) \)
- \( (f \cdot g)(x) \) or \( (f \times g)(x) = f(x) \cdot g(x) \)
- \( (f \div g)(x) \) or \( (f/g)(x) = \frac{f(x)}{g(x)} \), only where \( g(x) \neq 0 \)
- \( (f \circ g)(x) \) or \( f(g(x)) \): apply \( g \) first, then \( f \) (composition)
- \( \text{Domain}(f \circ g) = \{ x \in \text{Domain}(g) : g(x) \in \text{Domain}(f) \} \)
Sample: f(x) = x+1 (blue), g(x) =
2-x (green)
Domains (Short Rules)
- \( \text{Domain}(f+g),\ (f-g),\ (f \cdot g) = \text{Domain}(f) \cap \text{Domain}(g) \)
- \( \text{Domain}(f \div g) = \text{Domain}(f) \cap \text{Domain}(g) \), excluding \( x \) where \( g(x) = 0 \)
- \( \text{Domain}(f \circ g) = \{ x \in \text{Domain}(g) : g(x) \in \text{Domain}(f) \} \)
How to Graph a Combined Function
- Pick \( x \)-values covering key features (intercepts, turning points, asymptotes).
- Make a table: \( x \) | \( f(x) \) | \( g(x) \) | \( (f+g)(x) \) | \( (f-g)(x) \) | \( (f \cdot g)(x) \) | \( (f/g)(x) \) (if \( g(x) \neq 0 \)).
- Compute each result numerically and plot the points.
- Connect points smoothly, respecting continuity and asymptotes.
Notes:
- \( (f+g) \): add y-values; zeros where \( f(x) + g(x) = 0 \).
- \( (f-g) \): subtract y-values; zeros where \( f(x) = g(x) \).
- \( (f \cdot g) \): zeros are any zeros of \( f \) or \( g \); sign = sign(f)×sign(g).
- \( (f/g) \): zeros are zeros of \( f \) (if \( g \neq 0 \)); vertical asymptotes where \( g = 0 \); holes if both \( f \) and \( g \) are zero at same \( x \).
Sample: f(x) = x+1 (blue), g(x) =
2-x (green), f+g (red)
Product, Quotient, and Composition — Visuals
Sample: f(x) = x (blue), g(x) = 2-x
(green), f·g (purple), f/g (red, vertical asymptote)
Sample: g(x) = x² (green), f∘g =
2x²+1 (blue)
Quick Evaluation Examples (Algebraic)
- Let \( f(x) = x^2 + 1 \), \( g(x) = 3x \):
- \( (f + g)(1) = 2 + 3 = 5 \)
- \( (f - g)(2) = 5 - 6 = -1 \)
- \( (f \cdot g)(0) = 1 \cdot 0 = 0 \)
- \( (f \div g)(1) = 2/3 \) (only if \( g(1) \neq 0 \))
- With graphs: read \( f(x) \) and \( g(x) \) at that \( x \), then compute.
Composition — Forming and Evaluating f∘g and g∘f
- Definition: \( (f \circ g)(x) = f(g(x)) \). Evaluate by two steps: first compute \( y = g(x) \), then \( f(y) \).
- Example: \( f(x) = 2x+1,\ g(x) = x^2 \)
- \( (f \circ g)(x) = 2x^2 + 1 \)
- \( (g \circ f)(x) = (2x+1)^2 = 4x^2 + 4x + 1 \)
- \( (f \circ g)(4) = 33 \), \( (g \circ f)(5) = 121 \)
- Domain caution: \( f(x) = \sqrt{x} \) (domain \( x \geq 0 \)), \( g(x) = x-3 \).
\( (f \circ g)(x) = \sqrt{x-3} \), domain \( x \geq 3 \).
\( (g \circ f)(x) = \sqrt{x} - 3 \), domain \( x \geq 0 \).
Evaluating Compositions from Graphs
- To find \( (f \circ g)(a) \):
- Read \( g(a) \) from g-graph (call this \( y \)).
- If \( y \) not in domain of \( f \), then \( (f \circ g)(a) \) is undefined.
- Otherwise, read \( f(y) \) from f-graph; that is \( (f \circ g)(a) \).
- Example: if \( g(-2) = 3 \) and \( f(3) = -1 \), then \( (f \circ g)(-2) = -1 \).
Given h(x): Strategies to Propose f and g
- If \( h = f + g \): look at \( h(0) \) and zeros to split into two simple summands.
- If \( h = f \cdot g \): zeros of \( h \) must be zeros of \( f \) or \( g \); factor \( h \) and assign factors to \( f \) or \( g \).
- If \( h = f \div g \): vertical asymptotes indicate zeros of \( g \); zeros of \( h \) indicate zeros of \( f \) (if numerator not zero).
- If \( h = f \circ g \): look for inner mapping behaviour — repeated patterns or restricted ranges hint at the inner \( g \).
- Decompositions are not unique; choose simple, justifiable \( f \) and \( g \) matching intercepts, zeros, and asymptotes.