Lesson 5: Function Operations

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 functions $f(x)$ and $g(x)$, where $g(x) \neq 0$, the division of the two functions is defined as:

Examples

• For functions $f(x) = x^2 - 6x - 16$ and $g(x) = x + 2$, find $(\frac{f}{g})(x)$. We calculate $\frac{x^2 - 6x - 16}{x + 2}$. By factoring the numerator to $(x-8)(x+2)$, we simplify to get $(\frac{f}{g})(x) = x - 8$.
• Using the functions from the previous example, find $(\frac{f}{g})(10)$. Substitute $x=10$ into the simplified result: $(\frac{f}{g})(10) = 10 - 8 = 2$.
• For $f(x) = x^3 + 1$ and $g(x) = x + 1$, find $(\frac{f}{g})(x)$. We compute $\frac{x^3+1}{x+1}$. Using polynomial division, the result is $x^2 - x + 1$.

Explanation

The notation $(\frac{f}{g})(x)$ is simply a formal way to express the division of one polynomial function, $f(x)$, by another, $g(x)$. To solve, you set up the division as a fraction and use a method like long division or factoring to find the result.