Multiply and divide functions and analyze domains. Math 30-1 Alberta.
Lesson 1.4 of Function Transformations & Operations in Math 30-1 — Alberta curriculum lessons.
The Big Idea. Just like we can add and subtract functions, we can also multiply them!
Product of Functions. A combined function of the form h(x) = f(x) · g(x) represents the product of two component functions, f(x) and g(x).
Product of Functions
(f · g)(x) = f(x) · g(x)
Important Domain Rule. Before we work with products, we need to understand how their domains work.
For Products (f · g)(x):
The domain includes all x-values where both f(x) and g(x) are defined. If either function is undefined at some x-value, the product is also undefined there.
Key Point: The domain of (f · g) is the intersection of the domains of f and g.
Multiplying functions multiplies their outputs at each x-value.
If f(x) is large and g(x) is large, then the product becomes very large. If either function equals zero at some x-value, the product equals zero at that x-value. Key Insight: Zeros of either function become zeros of the product. This is important for understanding polynomial behavior later!
Problem. Given: f(x) = 2x + 1 and g(x) = x - 4
Find: (f · g)(x)
Step 1: Write the definition
(f · g)(x) = f(x) · g(x)
Step 2: Substitute
(f · g)(x) = (2x + 1)(x - 4)
Step 3: Expand using FOIL
(f · g)(x) = (2x + 1)(x - 4)
FOIL Method. First terms: 2x · x = 2x^2
Outer terms: 2x · (-4) = -8x
Inner terms: 1 · x = x
Last terms: 1 · (-4) = -4
Add all terms together:
(f · g)(x) = 2x^2 - 8x + x - 4
Step 4: Combine like terms
(f · g)(x) = 2x^2 - 7x - 4
Domain. Since both f(x) and g(x) are polynomials, they are defined for all real numbers.
Domain: \x R\ or (-∞, ∞)
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