Lesson 6: Operations With Functions

Property

For functions $f(x)$ and $g(x)$,

Examples

• Let $f(x) = 2x+3$ and $g(x) = x^2-1$. The sum is $(f+g)(x) = (2x+3) + (x^2-1) = x^2 + 2x + 2$.

• Let $f(x) = 4x^2 - 5$ and $g(x) = x^2 + 2x$. The difference is $(f-g)(x) = (4x^2 - 5) - (x^2 + 2x) = 3x^2 - 2x - 5$.

Property

To find the product of two functions, $(fg)(x)$, multiply their polynomial expressions using the Distributive Property (or the vertical multiplication method).

When you multiply two non-zero polynomials, the degree (highest exponent) of the new combined function is exactly the sum of the degrees of the original functions.

Examples

• Binomial times Trinomial: Let $f(x) = x + 2$ and $g(x) = x^2 + 3x + 1$. Find $(fg)(x)$.

Distribute the $x$: $x(x^2 + 3x + 1) = x^3 + 3x^2 + x$
Distribute the 2: $2(x^2 + 3x + 1) = 2x^2 + 6x + 2$
Combine like terms: $x^3 + 5x^2 + 7x + 2$

• Predicting the Degree: Let $P(x) = 3x^5 + x$ (degree is 5) and $Q(x) = 2x^2 - 1$ (degree is 2).

Without doing the full multiplication, we know the degree of the product $(PQ)(x)$ will be $5 + 2 = 7$.

Explanation

Property

For combined functions $(f \pm g)(x)$ or $(f \cdot g)(x)$, the domain is the intersection of the domains of $f(x)$ and $g(x)$. The range must be determined by analyzing the behavior of the combined function.

Examples