Functions

 

I. Functions

A relation is any set of ordered pairs, (x,y). The collection of all possible x-values in the ordered pairs is called the domain. The set of all possible y-values in the ordered pairs is called the range. Note that each value in the domain is also known as an input value, or independent variable, and is often labeled with the lowercase letter 𝑥. Each value in the range is also known as an output value, or dependent variable, and is often labeled lowercase letter 𝑦.

 

A function is a specific type of relation in which each input value has one and only one output value. A function can be thought of as sort of mathematical machine. You enter the input, the machine follows a specific rule, and produces the output. 

Function Notation

The notation 𝑦=𝑓⁡(𝑥) defines a function named 𝑓. This is read as ‶𝑦 is a function of 𝑥″  or "y equals f of x". The letter 𝑥 represents the input value, or independent variable. The letter 𝑦, or 𝑓⁡(𝑥), represents the output value, or dependent variable. Note that these are the most common notations used but other letters can also be used. 

To read function notation out loud, state the letter representing the function name followed by the word "of" and the input value, such as pronouncing ƒ(x) as "f of x."

ƒ(x) = 4x +1  read as "f of x equals 4x plus 1.

g(t) = 3t  read as "g of t equals 3t

II. Solving Function Problems

To solve a function problem, substitute the given input value for every instance of the variable in the equation and simplify the resulting expression to determine the output.

Example 1:

ƒ(x) = x + 5

Find the value of ƒ(10)

Here, the rule is add 5. Solve by simply replacing the x with the input value 10.

ƒ(10) = 10 + 5 = 15

III. Obtaining a Function from an Equation

1. The Conceptual Difference
Before you start the algebra, it is important to understand what you are trying to achieve:

• An Equation is a broad statement of equality between two expressions (e.g., x² + y² = 25). It describes a relationship, but one input (x) might lead to multiple outputs (y).
• A Function is a specific type of equation where every value of x (the input) is paired with exactly one value of y (the output).

2. Step-by-Step: Obtaining the Function

Step 1: Isolate the Dependent Variable
In most mathematics, y is the dependent variable. To find the function, you must solve the equation for y so that it stands alone on one side of the equals sign.

Example:
Start with the equation: 2x + y = 8

Subtract 2x from both sides:

y = -2x + 8

Step 2: Test for "Function-ness"
Once y is isolated, you must ensure that a single x value cannot produce two different y values. This is often checked using the Vertical Line Test if you are looking at a graph, or by checking for ± symbols in your algebra.

If you solve for y and end up with y = ±√x, the equation is not a function because one x (like 4) would result in two y values (2 and -2).

Step 3: Use Function Notation
If the equation passes the test, replace the variable y with the function notation ƒ(x). This signals that the value of the output depends directly on the input x.

Result:
ƒ(x) = -2x + 8

Graphing Functions

See: Lumen: Graphing Linear Functions

Reference

OpenStax: College Algebra

Lumen: Elementary Algebra: Intro to Functions

Khan: Functions and linear models

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