Algebra Basics: What Are Functions? - Math Antics

Introduction

• Rob welcomes viewers to Math Antics and introduces the topic of functions.

"Hi, I'm Rob. Welcome to Math Antics!"

Functions in Math

• In math, a function is a way to relate or connect one set to another set.

• A set is a collection of things, which can be numbers, letters, or any other objects.

• Sets can be represented visually or using mathematical notation.

• Sets can have a finite or infinite number of elements.

"In math, a function is basically something that relates or connects one 'set' to another 'set' in a particular way. A set is just a group or collection of things. Often it's a collections of numbers, but it doesn't have to be."

Input and Output Sets

• Functions have an input set called the "domain" and an output set called the "range."

• Input values from the domain are related to output values in the range.

• Input and output sets are often represented in a function table.

"A function is something that takes each value from an input set and relates it (or maps it) to a value in an output set. The input set is usually called 'The Domain' and the output set is usually called 'The Range'."

Function Tables

• Function tables have two columns: one for input values and one for corresponding output values.

• The function itself is written above the function table in mathematical notation.

• Function tables help visualize the relation between input and output values.

"A function table normally has two columns: one on the left for the input values and one on the right for the corresponding output values."

Examples of Functions

• Examples of functions include relating polygon names to the number of sides and algebraic equations like y = 2x.

• Function tables can be used to show the input-output combinations.

• Patterns in functions can be observed, such as the relationship between input and output values.

"So a function doesn’t just relate a set of inputs to a set of outputs. A function relates a member of an input set to exactly one member of an output set."

Limitations of Functions

• Functions cannot have "one-to-many" relations, where one input value leads to multiple output values.

• The equation y squared equals x is an example of a relation that is not a function.

• Functions are defined by having only one output value for each input value.

"Functions aren't allowed to have what we call 'one-to-many' relations, where one particular input value could result in many different output values."

Graphing Functions

• Functions can be graphed on a coordinate plane.

• Different types of functions, such as linear, quadratic, and trigonometric functions, have distinct graphs.

• The Vertical Line Test is used to determine if a graph represents a function.

"You can GRAPH a function! Here are the points from our function table plotted on the coordinate plane, and here's the resulting graph we get if we connect those points."

Vertical Line Test

• The Vertical Line Test helps identify if a graph represents a function.

• A vertical line is moved across the graph, and if it intersects the graph at more than one point, the graph is not a function.

"The Vertical Line Test helps us see if a graph has any of those one-to-many relations that would disqualify it as a function."

Functions and Graphs

• Functions have one 'y' value for each 'x' value.

• If a graph passes the Vertical Line Test, it qualifies as a function.

"There’s only one ‘y’ value for each ‘x’ value, so the graph qualifies as a function."

Graphs that Don't Pass the Vertical Line Test

• The graph of the equation 'y squared' equals 'x' does not pass the Vertical Line Test.

• The domain of this equation does not include any negative input values.

• The graph intersects the vertical line at just one point for the positive inputs, but intersects at two points for the negative inputs.

• This equation is not considered a function because it gives two possible outputs for some of its inputs.

"But as we move to the right on the 'x' axis, you can see that our vertical line is now intersecting the curve in TWO places. That means this equation is giving us two possible outputs for some of its inputs, which means that it’s not considered a function."

Common Function Notation

• Functions are often written as 'f(x)' instead of 'y'.

• 'f(x)' represents a function that takes an input value 'x' and gives an output value 'y'.

• 'f(x)' is the name of the function, not a variable being multiplied.

• The notation 'f(x)' is used to highlight that you're dealing with a function with a specific input variable.

"Well, it turns out that a really common way to represent a function is this… This notation simply means that a function (named ‘f’) takes an input value (named ‘x’) and gives an output value (named ‘y’)."

Evaluating Functions

• The notation 'f(x)' can be used to evaluate functions for specific values.

• An example function is 'f(x) = 3x + 2'.

• To evaluate the function for the input value 4, you substitute 4 in place of any 'x's in the function.

• 'f(4) = 14' means the value of the function when the input is 4 is 14.

• This can be done for other values as well.

"Then you could then ask someone to evaluate the function for the input value 4 by saying what is f(4). That means you’ll substitute a 4 in place of any ‘x’s that are in the function. For this function, that would mean f(4) = 14."

Functions and Their Properties

• Functions relate an input value to exactly one output value.

• The set of all input values is called the domain, while the set of output values is called the range.

• Functions in algebra are typically equations that can be graphed on the coordinate plane.

• The input and output values are treated as ordered pairs for graphing.

"That’s what functions are in math. They’re things that relate an input value to exactly one output value. And the set of all input values is called the domain while the set of output values is usually called the range. In algebra, functions typically come in the form of equations that can be graphed on the coordinate plane by treating the input and output values as ordered pairs."