# Functions

The concept of a function is essential to the study of mathematics. A strong understanding of functions is necessary throughout math courses such as algebra, trigonometry, calculus, and beyond. Functions have a diverse range of applications in science, engineering, economics, medicine, and much more. For example, if a biologist wants to create a model of population growth, functions will come in handy for this project. In addition, if a civil engineer is designing a bridge and needs to make various calculations, an understanding of functions will be vital. The applications of function are wide-ranging, which is why it's a good idea to examine them closely. This will ensure that we have a solid mathematical foundation to build upon in the future. Before we look at functions, we might want to acquaint ourselves with a few other related topics, such as writing number patterns in function notation, the real zero of a function, and comparing functions.

## What is a function?

A function accepts an input, performs a mathematical operation on it, and then returns an output. Each input and output is called an ordered pair. The set of possible inputs, or x-values, is called the domain. The set of valid outputs, or y-values, is called the range. One important feature of a function is that it never has more than one output for every input. Each x-value must have one and only one y-value. If an input has more than one output, then it ceases to meet the definition of a function.

In a function, if we have an ordered pair of (1,3), we cannot also have an ordered pair of (1, 4). We can describe this in strictly mathematical terms by thinking of a function as a special relation where if (x_{1}, y) is in the same relation as (x_{2}, y), then x_{1} = x_{2}. One handy way to determine graphically if something meets the definition of a function is to perform the vertical line test. We must not be able to draw a vertical line through any point such that it intersects two areas on the graph. If we can, it's not a function.

Let's look at an example. Does the following set of ordered pairs describe a function?

{(1, 1), (1, 2), (2, 4), (3, 5)}

Since we have an x-value mapping onto two separate y-values, this set of ordered pairs does not describe a function.

Does the following set of ordered pairs describe a function?

{(-2, 4), (-1, 1), (0, 0), (1,1), (2, 4)}

Since we don't have an x-value mapping onto two separate y-values, this looks like a function. Notice that the x-values -1 and 1 pair with a single y-value of 1. This is okay. Only when a single x-value is paired with two distinct y-values is the definition of a function violated. It's a good idea to plot these points on graph paper and perform the vertical line test. This provides us with an extra tool for determining if a set of ordered pairs is a function. Also, it helps us visualize the problem on a graphical level.

We use a special notation when we write functions. The input of a function is called x. The output of a function is f(x). Note that we are not multiplying f by x. Rather, this notation indicates that we are inserting the variable x into the function. Let's look at an example problem.

Given the function: f(x) = x – 1

find the value of f(x) if x = 5.

For this problem, we simply insert 5 into the x-value.

f(5) = 5 – 1

f(5) = 4

We can even insert expressions into a function instead of mere numbers.

Given the function: f(x) = x^{2}

find the value of f(x+1).

f(x+1) = (x+1)^{2}

= (x+1)(x+1)

We use the FOIL method to expand this expression.

= x^{2} +1x+1x+1

= x^{2} +2x +1

## Function practice problems

1. Determine whether the following set of ordered pairs describes a function.

{(4,2), (1,1), (0,0), (1, -1), (4, -2)}

In this set of ordered pairs, the x-value 4 maps onto both y-values 2 and -2. Likewise, the x-value 1 maps onto both y-values 1 and -1. Therefore, this is not a function.

2. Given that: f(x) = ((x+1)/2) find f(3).

To solve this problem, we simply insert 3 into our function.

f(3) = ((3+1)/2)

= 4/2

= 2

3. Determine whether the following set of ordered pairs constitutes a function.

{(-2,1), (-1,2), (0,3), (1,4), (2,5)}

In this problem, each x-value maps onto its own separate y-value. Therefore, this is a function.

4. Given that: f(x) = x^{2} + 5 calculate f(7)

In this case, we plug 7 into f(x).

f(7) = 7^{2} + 5

= 49 + 5

= 54

5. Determine whether the following set of ordered pairs describes a function.

{(-2,-4), (-1,-1), (0,0), (1,-1), (2, -4)}

This is a function because no x-value shares two separate y-values.

6. Determine whether the following set of ordered pairs describes a function.

{(-4,2), (-1,1), (0,0), (-1,-1), (-4,-2)}

In this set, the x-value -4 shares two distinct y-values. Furthermore, the x-value -1 shares two separate y-values as well. This is not a function.

7. Given that: f(x) = 2x^{2} + 5x + 2 calculate f(x+3).

First, we plug x+3 into our function.

f(x+3) = 2(x+3)^{2} + 5(x+3) + 2

= 2(x+3)(x+3) + 5x + 15 + 2

= 2(x^{2} + 3x +3x + 9) + 5x +17

=2(x^{2} + 6x + 9) + 5x + 17

=2x^{2} + 12x + 18 + 5x + 17

= 2x^{2} + 17x + 35

8. Determine whether the following set of ordered pairs describes a function.

{(-2, -7), (-1,0), (0,1), (1,2), (2,9)}

In this set, no x-value pairs with multiple y-values. Therefore, it's a function.

## Topics related to the Functions

Writing Number Patterns in Function Notation

## Flashcards covering the Functions

## Practice tests covering the Functions

College Algebra Diagnostic Tests

## Get extra help with functions

Functions can be a difficult topic, especially if it's your first time studying them. If you're struggling, you may want to think about studying alongside a tutor. They will explain anything that's not clear and they can walk you through a series of practice problems. Further, your high school student can benefit from working with a tutor if they're having trouble in an Algebra I course. A tutor can develop a specialized learning plan to function as a roadmap for your student. If you have any questions or concerns, an Educational Director from Varsity Tutors will respond to them. They will get you or your student set up with a quality tutor as soon as you're ready to get started.

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