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# Functions

Functions are a mathematical relationship. They involve an input and produce an output. Using Algebra, functions can be written as f and the input as x, creating f(x). Functions can be complex and use different Algebra, for example,    orThere are two different types of functions, composite and inverse.

## Composite functions

A composite function involves combining two or more functions to create a new function. This is also known as a function of a function. For example, let's look at fg(x). This means that you first find g(x), then you use the output of that to solve f(x).

Given that f(x) = x + 2 and g(x) = 3x 1 find fg(4)

First, you need to solve g(4)

g(4) = 3(4) 1

g(4) = 11

Now you can put the output of g(4), which was 11, into your f function to find fg(4)

f(11) = 11 + 2

f(11) = 13

Therefore fg(4) = 13

It is important to solve the functions in a specific order as fg(x) is not the same as gf(x). Let's have a look at solving gf(4) to see how the answer is different:

Given that f(x) = x + 2 and g(x) = 3x − 1 find gf (4)

This time you need to solve f(4) first

f(4) = 4 + 2

f(4) = 6

Now you can use that output to find g(x) using 6

g(6) = 3(6) -1

g(6) = 17

Therefore gf(4) = 17. Remember, solve the function that is closest to the brackets first.

## Inverse functions

An inverse function is when the function takes the opposite operation to the original function. It is shown as. The function takes the outputs and maps them back to the input, and this means that Inverse functions can only be mapped as one to one. If we plot Inverse functions on a graph, the line of the graph ofand will reflect each other.

Consider f(x) = 2x + 4

Let f(x) = 2x + 4 = y

y = 2x + 4

$x=\frac{y-4}{2}={f}^{-1}\left(y\right)$

This is the inverse of f(x).

## What are mappings?

Mapping can take an input from a set of numbers and transform it into an output. A mapping can be considered as a function if an input creates a distinct output. Below are the four ways that we can map inputs and outputs:

Mapping inputs and outputs

Only two of these mappings create functions; they are one to one and many to one. The terms domain and range can be used when discussing input and output:

• Domain is known as the possible inputs for the mapping

• Range is all of the possible outputs for the mapping

## How are graphs used for functions?

Graphs are able to give you a visual representation of a function, each function will give you a different type of graph. There are many different Factors that will change the way the graph looks, such as;

• If the function is negative or positive.

• The equation of the function.

### Polynomial graphs

Polynomials can be described as expressions that may contain variables that are raised to a positive power, which may also be multiplied by a coefficient. Polynomials can seem complicated but they can also look very simple, for example, $4{x}^{3}+3{x}^{2}+2x+x$is a polynomial but so is$2x+3$. These expressions are also graphed to give you a visual representation and just like Graphs of functions they can look very different depending on the polynomial that is being graphed.

## What are inequalities?

Inequalities are algebraic expressions that show how one term is less than, greater than or equal to another term. The symbols used to represent this are;

• $>$ Greater than

• $<$ Less than

• $\ge$ Greater than or equal to

• $\le$ Less than or equal to

$2x>4$

This shows you that 2x is greater than 4

$x<10$

This shows you that x is less than 10

$2{x}^{3}+5\ge 20$

This shows you that $2{x}^{3}$+ 5 is greater than or equal to 20

## Functions - Key takeaways

• Functions have an input that affects the output.

• Functions can be written using algebra.

• There are two different types of functions, composite and inverse.

• Mapping is used to show the domain and range of a function.

(explanation) en-pure maths-functions-mappings

###### Learn with 0 Functions flashcards in the free StudySmarter app

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Why are functions important?

Functions can be used in many different real-life situations.

How do you find a function?

You can find a function using graphs or mappings, you know that if you are given a mapping, it is a function if it is one to one or many to one.

What are functions?

Functions are a mathematical relationship involving inputs and outputs.

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