2.7 Combining Functions
Ideas in this lesson
- Combine functions algebraically and preserve domains
- Read graphs and construct graphical sums
- Evaluate ordered and repeated compositions
- Compose functions from tables and graphs
- Find composite domains before simplifying
- Build and decompose two- and three-stage functions
- Model geometric and motion chains
- Model revenue, profit, and ordered discounts
- Analyze linear composition and recover unknown functions
- Justify iteration and parity of compositions
Combine functions algebraically and preserve domains · Worked example
For f(x)=\(x \left(x + 4\right)\) and g(x)=\(\left(x - 4\right) \left(x + 4\right)\), find f+g, f−g, fg, and f/g, with all four domains. Preserve any excluded input after simplifying.
Answer: f+g: \(2 \left(x - 2\right) \left(x + 4\right)\); f−g: \(4 \left(x + 4\right)\); fg: \(x \left(x - 4\right) \left(x + 4\right)^{2}\); f/g: \(\dfrac{x}{x - 4}\). All real inputs for sum, difference and product; quotient excludes -4, 4.
Add and subtract like terms; multiply the two original expressions to obtain f+g: \(2 \left(x - 2\right) \left(x + 4\right)\); f−g: \(4 \left(x + 4\right)\); fg: \(x \left(x - 4\right) \left(x + 4\right)^{2}\).
Divide the original expressions and simplify to \(\dfrac{x}{x - 4}\).
The original denominator vanishes at -4, 4. All real inputs for sum, difference and product; quotient excludes -4, 4.
Divide the original expressions and simplify to \(\dfrac{x}{x - 4}\).
The original denominator vanishes at -4, 4. All real inputs for sum, difference and product; quotient excludes -4, 4.