# Function Basics

A function is a mathematical expression that relates inputs to their respective outputs. It can be thought of as a machine. These machines are fed something called an input, and produce something different, called an output.

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## Function basics definition

A function is a relation that associates each value of the departing set, called input, with one single value, called output, of the arrival set.

### Notation of a function

A function is denoted usually by $f\left(x\right)$, where the input is $x$ and the output is $f\left(x\right)$.

### Basic Attributes of functions

There are several key attributes that are helpful in determining whether a mathematical relation is a function. These attributes serve an important role in fully defining any function.

#### Input

Input is the independent value that is fed into a function, to produce an output.

The input of a function is a placeholder, that can take any value of the departing set to produce one output.

#### Output

Output is the value of the function arising from a certain input. The output is referred to as the dependent variable since its value depends on the input value.

A function to describe the projected growth in a population of rabbits in a field over time could have an output representative of the number of rabbits in the field at a given point in time.

A simple way to visualize this relationship between inputs and outputs in functions is the function diagram.

Below shows a diagram representing inputs and outputs for the function, $f\left(x\right)=2x.$

Inputs relate to outputs via functions- StudySmarter Originals

The below function is written in terms of a variable, x. Notice the function notation, the x inside the brackets indicates that this is a function with x as an input.

$f\left(x\right)=3x+8$.

Now, how do we actually use this function? Well, we want to find the output of this function for a given input, so it is simply a case of replacing any x in the function with said input.

Let's try this for an input of 10. All we have to do is replace every x with a 10.

$f\left(10\right)=3×10+8=30+8=38.$

So for an input of 10, we have an input of 38.

Let's try another input value, for example $x=0.$In this case, the output value would be $f\left(0\right)=3×0+8=0+8=8.$

 Function Not a function $f\left(x\right)=2x+3$ $f\left(b\right)=\frac{±\sqrt{{b}^{2}+4}}{2}$ $f\left(t\right)={t}^{3}+t+1$ ${x}^{2}={y}^{2}+1$

We consider the above table where we distinguish between functions and non functions for the following reasons:

• For $f\left(x\right)=2x+3$, for every input x, we have one single output f(x).
• The same goes for $f\left(t\right)={t}^{3}+t+1$, for every input t, we have one single output f(t).
• However, for $f\left(b\right)=\frac{±\sqrt{{b}^{2}+4}}{2}$, each input b, has 2 different outputs f(b), namely the positive and the negative of the fraction in question.
• The same goes for ${x}^{2}={y}^{2}+1$, while rearranging so we have y as the subject we get, $y=±\sqrt{{x}^{2}-1}$, and hence for every input x, we have two different outputs, namely the positive and the negative root of ${x}^{2}-1$.

#### Domain

The domain of a function is the set of all possible inputs of a function.

The domain is an important function attribute as it describes what input values are acceptable inputs to the function. It is used to avoid any undefined or undesirable outputs.

For instance, we take the following function

$f\left(x\right)=\frac{15}{x-7}$

The input variable of this function is x, so for what values of x does there not exist an output?

Well, we know that the bottom half of the fraction cannot be zero as dividing by zero is impossible. Now we just have to find the value of x at which the bottom of the function is equal to zero and we can include it in our domain.

$x-7=0$

$x=7$

So, for our function to be defined, our independent variable x can take any values in the set of real numbers, except $x=7.$Thus, we denote the domain of our function as follows

$Domai{n}_{f}={D}_{f}=\left\{x:x\in \mathrm{ℝ}\overline{)x\ne }7\right\}$.

This means that the domain of the function f(x) is simply the set of all real numbers, apart from 7. In essence, for any real number input apart from 7, the function has a unique, and real output.

Determine the domain of the following function,

$h\left(x\right)=\sqrt{x+9}$

Solution

As the square root of a negative does not exist, the value of the function inside the square root must not be negative. So we should have

$x+9\ge 0$

Hence, we can deduce that

$x\ge -9$

And from this, we can define the domain of this function as the set of all real numbers greater than -9,

${D}_{h}=\left\{x:x\in \mathrm{ℝ}\overline{)x\ge -9}\right\}$

#### Range

The range of a function represents all possible output values of a function.

Let's take a look at the same function as before, f(x).

$f\left(x\right)=\frac{15}{x-7}$

We have already found its domain,

${D}_{f}=\left\{x:x\in \mathrm{ℝ}\overline{)x\ne }7\right\}$

What can the output of f(x) not be? Well, no matter what value of x we take as our input to the function, it is impossible for f(x) to be equal to zero. The value of f(x) can get infinitely closer to zero for larger and larger values of x, but it will never actually reach it.

Thus, we can deduce that the range of our function is the set of all real numbers except for zero.

$Rangeoff=\left\{z:z\left(x\right)\in \mathrm{ℝ}\overline{)z\left(x\right)\ne 0\right\}=\mathrm{ℝ}\\left\{0\right\}.}$

Determine the range of the following function,

$r\left(t\right)=\sqrt{2t}$

Solution

Well, as it is impossible for the square root of a number to be negative, finding the range of this function is relatively direct.

The output of r(t) must be a positive number, and so the range of r(t) is the set of all real, positive numbers and zero,

$Rangeofr\left(t\right)=\left\{r\left(t\right)\ge 0\right\}={\mathrm{ℝ}}^{+}$

## Evaluating functions

Evaluating a function means finding the output that corresponds to a given input.

This is usually done through a direct substitution of the input value and a simplification of the right-side of the expression. Let's look at this in practice.

Evaluate the below function when x=4.

$f\left(x\right)={x}^{2}+12$

Solution

Finding the solution to the function when x = 4 is as simple as substituting 4 in place of each x.

$f\left(4\right)={4}^{2}+12$

Now, solving the function is just as simple as solving any other equation,

$f\left(4\right)=16+12=28$

Let's have a look at another, just to make sure we've got it.

Consider the function h(t). Find the solution to the function h(t) when t = 13.

$h\left(t\right)=\frac{18}{t+15}$

Let's sub in our given value of t,

$h\left(13\right)=\frac{18}{13+15}$

And simplify the right-hand side of the equation, to get

$h\left(13\right)=\frac{18}{28}=\frac{9}{14}$

## Graphing functions

One very powerful way to represent functions is with graphs. Graphs provide a great visual representation, that can be used to interpret functions and discern their properties.

Graphs of functions are simply comprised of the function's outputs on one axis, (typically the vertical axis) and the function's inputs on the other axis (typically the horizontal axis). Below are some examples of various functions and their graphs.

$f\left(x\right)=2x+5$

Graph of a straight line function - StudySmarter Originals

$h\left(t\right)={t}^{2}-3$

Graph of quadratic function - StudySmarter Originals

How exactly is it that we can graph function like this for ourselves? Well, doing this by hand is a simple case of plotting a series of individual points on a set of axes, and joining them up with a smooth line.

This can be done while following the steps.

1. Take a large set of input values from the domain of the function.
2. Calculate the corresponding outputs.
3. Plots the couple of points we already found in steps 1 and 2 into the graph.
4. Join the points with a smooth curve.

Let's take a look at a quick example to see how we do this.

On a set of axes, plot the graph of the function $f\left(x\right)={x}^{2}-2$.

Solution

The first step in plotting the function is to find the domain of the function.

Since f is a polynomial function, its domain is the set of all real numbers.

Let's take the inputs $x=\left\{-3,-2,-1,0,1,2,3\right\}$. We can find their corresponding outputs by substituting them through the function.

We recall that $f\left(x\right)={x}^{2}-2$, thus

 x f(x) x=-3 $f\left(-3\right)={\left(-3\right)}^{2}-2=7$ x=-2 $f\left(-2\right)={\left(-2\right)}^{2}-2=2$ x=-1 $f\left(-1\right)={\left(-1\right)}^{2}-2=-1$ x=0 $f\left(0\right)={0}^{2}-2=-2$ x=1 $f\left(1\right)={1}^{2}-2=-1$ x=2 $f\left(2\right)={2}^{2}-2=2$ x=3 $f\left(3\right)={3}^{2}-2=7$

This gives us the following points to be plotted on the graph,

$\left(-3,7\right),\left(-2,2\right),\left(-1,-1\right),\left(0,-2\right),\left(1,-1\right),\left(2,2\right),\left(3,7\right)$

Next, we plot these points on the graph.

Plotting points of the example function- StudySmarter Originals

The final stage is to join up the dots with a smooth curved line.

Graphing a function - StudySmarter Originals

Now, what if we wanted to find the value of one of these functions for a given input? Well, let's try and find the output of h(t) when t = 3.

All we have to do is find the point where the line t = 3 intercepts with the function h(t).

$h\left(t\right)={t}^{2}-3$

Finding the output of a function for a given input - StudySmarter Originals

And so, from the graph, we can read that the solution to the function h(t) when t = 3 is 6.

Not only can we use a vertical line to find the output of a function, but can also be used to test if a given graph is of a function at all. Let's take a look at how!

### Vertical Line Test

As previously discussed, for a given input, a function can only have one single output. Due to this attribute, there is a simple test that can be done to check if a graph is of a function.

Put simply, if a graph is of a function, any vertical line drawn will only cross it once. If a vertical line can be drawn that crosses the graph more than once, then the graph is not of a function.

Below is an example of the vertical line test used on a function.

Vertical line test on a function example - StudySmarter Originals

Though this graph below may seem like a function at first glance, we can see from inspection and the vertical-line test that each input has two outputs, therefore the below graph is not a function.

Vertical line test conducted on a non-function example - StudySmarter Originals

## Basic Types of Functions

There are a few basic types of functions that it is important to be able to recognize. Let's take a look at a few type of functions sand their corresponding graphs.

Linear function.

Linear functions can always be expressed in the form $y=ax+b$, for instance,$y=2x+3$.

The graph of a linear function is a straight line.

Graph of a linear function- StudySmarter Originals

Quadratic functions are of the form $f\left(x\right)=a{x}^{2}+bx+c.$ For example,$f\left(x\right)=3{x}^{2}+2x+5$.

The graph of a quadratic function is a parabola, a U shaped curve opening upwards or downwards.

Graph of a quadratic function - StudySmarter Originals

Trigonometric functions.

Trigonometric functions such as f(x)=sin x, f(x)=cos x and f(x)=tan x. whose graphs are plotted below.

$f\left(x\right)=\mathrm{sin}\left(x\right)$

Graph of a sine function- StudySmarter Originals

$f\left(x\right)=\mathrm{cos}\left(x\right)$

Graph of a Cosine function - StudySmarter Originals

$f\left(x\right)=\mathrm{tan}\left(x\right)$

Graph of a tangent function - StudySmarter Originals

There are so many more types of functions that we haven't mentioned. There are logarithmic functions, exponential functions, cubic functions, quartic functions and so many more! Why not head over to our article on Types of Functions to find out more!

## Examples of functions

We finalize our article with some simple yet incredibly complicated graphs of functions.

Here is an example of a very simple linear function.

$f\left(x\right)=3x+2$

Example of a linear function - StudySmarter Originals

We can get more complicated functions, such as this example of a quadratic function.

$f\left(x\right)=4{\left(x-3\right)}^{2}+2$

Example of a quadratic function - StudySmarter Originals

Functions can even be as crazy as the one below. There really are unlimited possibilities!

$f\left(x\right)=\frac{2{x}^{2}+4{x}^{3}-3x+{3}^{x}}{2x+7}-\frac{{4}^{x}-{3}^{x}+\mathrm{ln}\left(x\right)}{5}$

Example of a complicated function - StudySmarter Originals

So there we have it, those are the basics of functions! All of this is just the tip of the iceberg though, why not check out some of our other explanations on various aspects of functions to uncover the rest!

## Function Basics - Key takeaways

• Functions are mathematical expressions that relate inputs to outputs.
• By definition, each input to a function can only have a single output.
• The domain of a function is the set of all possible inputs of a function.
• Solving a function for a given input is simply a case of replacing the function variable with the value of said input, and solving the right-hand side of the equation.

#### Flashcards in Function Basics 37

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What is a function in algebra?

A function is a mathematical expression which has no more than one output for a given input.

What are types of functions?

Some types of functions are trigonometric functions, linear functions, quadratic functions, cubic functions and logarithmic functions.

What equations are functions?

Equations are functions if they have no more than a single output for a given input.

What is the most basic function in math?

The simplest function in maths could arguably be any kind of constant function, i.e. y = 5. For constant functions such as this, the output for all inputs is the same, in this case, 5.

What are examples of functions?

Some examples of functions are f(x) = sin(x), f(x) = ln(x), and f(x) = 2x + 1.

Each of these functions has no more than a single output for a given input.

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