Learn on PengiAoPS: Introduction to Algebra (AMC 8 & 10)Chapter 16: Functions

Lesson 16.2: Combining Functions

In this Grade 4 AMC math lesson from AoPS: Introduction to Algebra, students learn how to combine two functions through addition, subtraction, multiplication, and division to create new functions. The lesson covers how the domain of a combined function is determined by the domains of the original functions, with special attention to quotient functions where the denominator cannot equal zero. Students work through problems using concrete examples like linear and radical functions to build and verify these combined function rules.

Section 1

Add and Subtract Polynomial Functions

Property

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

(f + g)(x) = f(x) + g(x)

(f - g)(x) = f(x) - 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 $(f-g)(x) = (4x^2 - 5) - (x^2 + 2x) = 3x^2 - 2x - 5$ .

Section 2

Multiplying Functions and Degree Behavior

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

Section 3

Division of Polynomial Functions

Property

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

\left(\dfrac{f}{g}\right)(x) = \dfrac{f(x)}{g(x)}

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.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Add and Subtract Polynomial Functions

Property

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

(f + g)(x) = f(x) + g(x)

(f - g)(x) = f(x) - 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 $(f-g)(x) = (4x^2 - 5) - (x^2 + 2x) = 3x^2 - 2x - 5$ .

Section 2

Multiplying Functions and Degree Behavior

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

Section 3

Division of Polynomial Functions

Property

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

\left(\dfrac{f}{g}\right)(x) = \dfrac{f(x)}{g(x)}

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

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Add and Subtract Polynomial Functions

Property

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

(f + g)(x) = f(x) + g(x)

(f - g)(x) = f(x) - 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 $(f-g)(x) = (4x^2 - 5) - (x^2 + 2x) = 3x^2 - 2x - 5$ .

Section 2

Multiplying Functions and Degree Behavior

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

Section 3

Division of Polynomial Functions

Property

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

\left(\dfrac{f}{g}\right)(x) = \dfrac{f(x)}{g(x)}

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 $x^2 - x + 1$ .