Learn on PengienVision, Algebra 1Chapter 10: Working With Functions

Lesson 6: Operations With Functions

In this Grade 11 enVision Algebra 1 lesson, students learn how to add, subtract, and multiply functions by applying the operations (f + g)(x) = f(x) + g(x), (g − f)(x) = g(x) − f(x), and (f · g)(x) = f(x) · g(x). The lesson also explores how combining functions affects their domain and range, including cases where a linear and quadratic function are combined to produce a new quadratic function. A real-world application using cylinder surface area shows how function operations model practical geometric problems.

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

Domain and Range of Combined Functions

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

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

Domain and Range of Combined Functions

Property

Examples

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 $5 + 2 = 7$ .