Learn on PengienVision, Algebra 1Chapter 3: Linear Functions

Lesson 2: Linear Functions

In this Grade 11 enVision Algebra 1 lesson, students learn to identify, evaluate, graph, and write linear functions using function notation, slope-intercept form, and constant rates of change. The lesson covers how to express equations like f(x) = 15x + 2 in function notation and apply linear functions to real-world scenarios such as calculating costs or modeling temperature change. Students also analyze the domain and range of linear functions within the context of practical situations.

Section 1

Function Notation

Property

For the function $y = f(x)$ :

f \quad \text{is the name of the function}

x \quad \text{is the domain value}

f(x) \quad \text{is the range value} \quad y \quad \text{corresponding to the value} \quad x

We read $f(x)$ as $f$ of $x$ or the value of $f$ at $x$ . The process of finding the value of $f(x)$ for a given value of $x$ is called evaluating the function.

Examples

For the function $f(x) = 5x - 4$ , to evaluate $f(3)$ , we substitute 3 for $x$ : $f(3) = 5(3) - 4 = 15 - 4 = 11$ .

For the function $g(x) = x^2 + 2x$ , to evaluate $g(a)$ , we substitute $a$ for $x$ : $g(a) = a^2 + 2a$ .

Section 2

Converting Between Function and Equation Notation

Property

Linear functions can be written in two equivalent forms: function notation $f(x) = mx + b$ and equation notation $y = mx + b$ .
To convert between them, replace $f(x)$ with $y$ or replace $y$ with $f(x)$ .

Examples

Section 3

Evaluating a Function

Property

Finding the value of the output variable that corresponds to a particular value of the input variable is called evaluating the function.
To evaluate a function described by an equation, we substitute the given input value into the equation to find the corresponding output, or function value.

Examples

Given the function $f(x) = 5x - 2$ , to evaluate $f(3)$ , we substitute $x=3$ to get $f(3) = 5(3) - 2 = 15 - 2 = 13$ .

For the function $g(t) = t^2 + 10$ , evaluating at $t=-4$ means calculating $g(-4) = (-4)^2 + 10 = 16 + 10 = 26$ .

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Function Notation

Property

For the function $y = f(x)$ :

f \quad \text{is the name of the function}

x \quad \text{is the domain value}

f(x) \quad \text{is the range value} \quad y \quad \text{corresponding to the value} \quad x

We read $f(x)$ as $f$ of $x$ or the value of $f$ at $x$ . The process of finding the value of $f(x)$ for a given value of $x$ is called evaluating the function.

Examples

For the function $f(x) = 5x - 4$ , to evaluate $f(3)$ , we substitute 3 for $x$ : $f(3) = 5(3) - 4 = 15 - 4 = 11$ .

For the function $g(x) = x^2 + 2x$ , to evaluate $g(a)$ , we substitute $a$ for $x$ : $g(a) = a^2 + 2a$ .

Section 2

Converting Between Function and Equation Notation

Property

Examples

Section 3

Evaluating a Function

Property

Examples

Given the function $f(x) = 5x - 2$ , to evaluate $f(3)$ , we substitute $x=3$ to get $f(3) = 5(3) - 2 = 15 - 2 = 13$ .

For the function $g(t) = t^2 + 10$ , evaluating at $t=-4$ means calculating $g(-4) = (-4)^2 + 10 = 16 + 10 = 26$ .

Overview

Lesson overview

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

Overview

Section 1

Function Notation

Property

For the function $y = f(x)$ :

f \quad \text{is the name of the function}

x \quad \text{is the domain value}

f(x) \quad \text{is the range value} \quad y \quad \text{corresponding to the value} \quad x

We read $f(x)$ as $f$ of $x$ or the value of $f$ at $x$ . The process of finding the value of $f(x)$ for a given value of $x$ is called evaluating the function.

Examples

For the function $f(x) = 5x - 4$ , to evaluate $f(3)$ , we substitute 3 for $x$ : $f(3) = 5(3) - 4 = 15 - 4 = 11$ .

For the function $g(x) = x^2 + 2x$ , to evaluate $g(a)$ , we substitute $a$ for $x$ : $g(a) = a^2 + 2a$ .

Section 2

Converting Between Function and Equation Notation

Property

Examples

Section 3

Evaluating a Function

Property

Examples

Given the function $f(x) = 5x - 2$ , to evaluate $f(3)$ , we substitute $x=3$ to get $f(3) = 5(3) - 2 = 15 - 2 = 13$ .

For the function $g(t) = t^2 + 10$ , evaluating at $t=-4$ means calculating $g(-4) = (-4)^2 + 10 = 16 + 10 = 26$ .