Introduction to Functions: Fresh Take

  • Define what a function is and apply the vertical line test to identify functions
  • Use function notation to represent and evaluate functions
  • Recognize the graphs of fundamental toolkit functions

Determining Whether a Relation Represents a Function

The Main Idea 

A function is like a magical box that takes something in and spits something else out. In math terms, it’s a special relationship where each input (or “independent variable”) is connected to exactly one output (or “dependent variable”). So, if you feed the function a number, it will give you back another number, but it won’t ever get confused and give you two or more numbers for the same input.

  • Input: Also known as the independent variable, often labeled [latex]x[/latex].
  • Output: Also known as the dependent variable, often labeled [latex]y[/latex].
  • Domain: The set of all possible inputs.
  • Range: The set of all possible outputs.

How to: Given a relationship between two quantities, determine whether the relationship is a function.

  1. Identify the input values.
  2. Identify the output values.
  3. If each input value leads to only one output value, classify the relationship as a function. If any input value leads to two or more outputs, do not classify the relationship as a function.
The coffee shop menu, shown below, consists of items and their prices.

A menu of donut prices from a coffee shop where a plain donut is $1.49 and a jelly donut and chocolate donut are $1.99.
  1. Is price a function of the item?
  2. Is the item a function of the price?

Find the domain and range of the relation and determine whether it is a function.

[latex]\{(−3, 4),(−2, 4),( −1, 4),(2, 4),(3, 4)\}[/latex]

Verifying a Function Using the Vertical Line Test

The Main Idea 

The vertical line test is a quick way to determine if a graph represents a function. If any vertical line intersects the graph more than once, it’s not a function.

When you’re looking at a graph and wondering if it represents a function, try imagining or drawing vertical lines across the graph. If any of these lines touch the graph at more than one point, then what you’re looking at isn’t a function.

Does the graph below represent a function? Graph of absolute value function.

For more examples of the vertical line test, watch the following video.

You can view the transcript for “Ex 1: Use the Vertical Line Test to Determine if a Graph Represents a Function” here (opens in new window).

Determining Whether a Function is One-to-One

The Main Idea 

A function is one-to-one if each output corresponds to exactly one input. This ensures that the function has a unique inverse.

Think of a one-to-one function like a VIP list at a party. Each name on the list gets you one, and only one, ticket in. No name repeats, and no ticket can be used twice.

Is the area of a circle a function of its radius? If yes, is the function one-to-one?

For an overview of one-to-one functions, watch the following video.

You can view the transcript for “Overview of one to one functions” here (opens in new window).

The Horizontal Line Test

The Main Idea 

To check if a function is one-to-one, use the horizontal line test. If any horizontal line intersects the graph more than once, the function is not one-to-one.

Similar to the vertical line test, but this time you’re drawing horizontal lines. If any line touches the graph more than once, the function isn’t one-to-one, meaning it doesn’t have a unique inverse.

For more examples of the horizontal line test, watch the following video.

You can view the transcript for “Ex 1: Determine if the Graph of a Relation is a One-to-One Function” here (opens in new window).

Representing Functions Using Tables

The Main Idea 

How to: Given a table of input and output values, determine whether the table represents a function.

  1. Identify the input and output values.
  2. Check to see if each input value is paired with only one output value. If so, the table represents a function.

Function Notation

The Main Idea 

Function Notation: The function is often denoted as [latex]f(x)[/latex], where [latex]x[/latex] is the input and [latex]f(x)[/latex] is the output. This notation is a shorthand way of saying “the function [latex]f[/latex] acting on [latex]x[/latex] gives [latex]f(x)[/latex].”

Independent and Dependent Variables: In function notation, [latex]x[/latex] is the independent variable (the input), and [latex]f(x)[/latex] is the dependent variable (the output). The output depends on what you put in as the input.

Flexibility in Naming: Function notation allows you to use different names like [latex]f(x)[/latex], [latex]g(x)[/latex], or [latex]c(x)[/latex] to distinguish between multiple functions. Functions can have inputs that are not just numbers. For example, the input could be the name of a month, and the output could be the number of days in that month.

A function [latex]N=f\left(y\right)[/latex] gives the number of police officers, [latex]N[/latex], in a town in year [latex]y[/latex]. What does [latex]f\left(2005\right)=300[/latex] represent?

In the following videos we show two more examples of how to express a relationship using function notation.

You can view the transcript for “Ex: Function Notation Application Problem” here (opens in new window).


You can view the transcript for “Function Notation Application” here (opens in new window).

Finding Input and Output Values of a Function

Evaluation of Functions in Algebraic Forms

The Main Idea 

Evaluating Functions: To evaluate a function, substitute the given input into the function’s formula and perform the calculations. For example, if [latex]f(x)=x^2−8[/latex], and you want to find [latex]f(−3)[/latex], you’d calculate [latex](−3)^2−8=1[/latex].

Solving for Input: If you know the output and want to find the input, set the output equal to the function’s formula and solve for the input. For instance, if [latex]h(p)=p^2+2p[/latex] and [latex]h(p)=3[/latex], you’d solve the equation [latex]p^2+2p=3[/latex] to find the value of [latex]p[/latex].

Given the function [latex]h\left(p\right)={p}^{2}+2p[/latex], evaluate [latex]h\left(4\right)[/latex].

Given the function [latex]g\left(m\right)=\sqrt{m - 4}[/latex], evaluate [latex]g\left(5\right)[/latex].

Watch the video below for more examples of evaluating a function for specific values of the input.

You can view the transcript for “Evaluating Functions Using Function Notation (L9.3)” here (opens in new window).

The next video shows another example of how to solve a function.

You can view the transcript for “Ex: Find Function Inputs for a Given Quadratic Function Output” here (opens in new window).

Evaluating a Function Given in Tabular Form

The Main Idea 

Functions can be represented in tables, which can be particularly useful when equations are not the best fit.

The domain is the type of input, and the range is the output, often a real number.

To find the output for a given input, look for the input in the table and identify the corresponding output.

To find the input for a given output, scan the output column and note all instances of that output, then find the corresponding input(s).

Using the table below, evaluate [latex]g(1)[/latex].

[latex]n[/latex] [latex]1[/latex] [latex]2[/latex] [latex]3[/latex] [latex]4[/latex] [latex]5[/latex]
[latex]g(n)[/latex] [latex]8[/latex] [latex]6[/latex] [latex]7[/latex] [latex]6[/latex] [latex]8[/latex]

Watch the following video for more on evaluating a function given a table.

You can view the transcript for “Ex: Evaluate a Function and Solve for a Function Value Given a Table” here (opens in new window).

Finding Function Values from a Graph

The Main Idea 

Graphs can also be used to evaluate functions.

The graph represents a set of ordered pairs [latex](x,y)[/latex] or [latex](x,f(x))[/latex], where [latex]x[/latex] is the input and [latex]f(x)[/latex] is the output.

Using the graph, solve [latex]f\left(x\right)=1[/latex]. Graph of a positive parabola centered at (1, 0) with the labeled point (2, 1) where f(2) =1.

Watch the following video for more on finding function values given a graph.

You can view the transcript for “Ex 1: Determine a Function Value From a Graph” here (opens in new window).

Identifying Basic Toolkit Functions

The Main Idea 

In this section, we dive into the world of “Toolkit Functions.” Think of these as the ABCs of functions—your go-to set of basic functions that you’ll encounter throughout your studies. These toolkit functions come in various shapes and forms, from simple constant functions to more complex cube root functions. Knowing these functions by heart, including their graphs and equations, will give you a solid foundation for understanding more complex functions later on.

Toolkit Functions
Name Function Graph
Constant [latex]f\left(x\right)=c[/latex], where [latex]c[/latex] is a constant Graph of a constant function.
Identity/Linear [latex]f\left(x\right)=x[/latex] Graph of a straight line.
Absolute value [latex]f\left(x\right)=|x|[/latex] Graph of absolute function.
Quadratic [latex]f\left(x\right)={x}^{2}[/latex] Graph of a parabola.
Cubic [latex]f\left(x\right)={x}^{3}[/latex] Graph of f(x) = x^3.
Reciprocal [latex]f\left(x\right)=\frac{1}{x}[/latex] Graph of f(x)=1/x.
Reciprocal squared [latex]f\left(x\right)=\frac{1}{{x}^{2}}[/latex] Graph of f(x)=1/x^2.
Square root [latex]f\left(x\right)=\sqrt{x}[/latex] Graph of f(x)=sqrt(x).
Cube root [latex]f\left(x\right)=\sqrt[3]{x}[/latex] Graph of f(x)=x^(1/3).