A function is a combination of one or more mathematical operations that takes a set of numbers and changes them into another set of numbers .

Define a composite function .

A composite function is a function that 'combines' two functions. The output of one function is used as the input for the other function.

Give the definition of an inverse function .

An inverse function is a function that does the exact opposite operations of the function it came from.

True or false? Completing the square can help find inverse functions.

True. Completing the square can help rearrange a quadratic expression in order to find its inverse function.

GCSEMathsAQAHigherFlashcards2. AlgebraFunctions

Functions (AQA GCSE Maths)

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FrontIntroduction to Functions
What is a function?

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What is a function?
A function is a combination of one or more mathematical operations that takes a set of numbers and changes them into another set of numbers.
What does the notation $f (x) = . . .$ indicate?
The notation $f (x) = . . .$ indicates a function $f$ , where $x$ is the input and $f (x)$ is the output.
What is the input of a function?
The input of a function is the number being put into the function, often referred to as $x$ .
What is the output of a function?
The output of a function is the number coming out of the function, often referred to as $f (x)$ or $y$ .
How would you find the value of $f (3)$ for a function $f (x)$ ?
To find the value of $f (3)$ for a function $f (x)$ , substitute $x = 3$ into the equation of the function.
For example, if $f (x) = 2 x + 1$ , then $f (3) = 2 (3) + 1 = 6 + 1 = 7$ .
How would you find the value of $x$ if you knew that $f (x) = 15$ ?
To find the value of $x$ if you know that $f (x) = 15$ :
- Set the equation for $f (x)$ equal to 15
- Then solve for $x$
For example, if $f (x) = 2 x + 1$ and $f (x) = 15$ , then
$2 x + 1 = 15 \Rightarrow 2 x = 14 \Rightarrow x = 7$
Define a composite function.
A composite function is a function that 'combines' two functions.
The output of one function is used as the input for the other function.
What does the notation $fg (x)$ mean?
$fg (x)$ is the notation for a composite function made up of functions $f$ and $g$ .
It means " $f$ is applied to the output of $g (x)$ ".
I.e., first put $x$ into the function $g$ , and next put the output $g (x)$ into the function $f$ .
Note that the function on the right ( $g$ ) acts first, and the function on the left ( $f$ ) acts second.
What does the notation $gf (x)$ mean?
$gf (x)$ is the notation for a composite function made up of functions $g$ and $f$ .
It means " $g$ is applied to the output of $f (x)$ ".
I.e., first put $x$ into the function $f$ , and next put the output $f (x)$ into the function $g$ .
Note that the function on the right ( $f$ ) acts first, and the function on the left ( $g$ ) acts second.
How would you find the value of $fg (2)$ ?
To find the value of $fg (2)$ :
1. First substitute $x = 2$ into the equation for $g (x)$ .
2. Then substitute the output $g (2)$ into the equation for $f (x)$ .
Give the definition of an inverse function.
An inverse function is a function that does the exact opposite operations of the function it came from.
What is the notation for the inverse function of the function $f (x)$ ?
The notation for the inverse function of the function $f (x)$ is $f^{- 1} (x)$ .
What are the steps to find the inverse function $f^{- 1} (x)$ of $f (x)$ ?
To find $f^{- 1} (x)$ :
1. Write $f (x)$ as $y = (formula in x)$ .
2. Swap $x$ and $y$ to get $x = (formula in y)$ .
3. Rearrange to make $y$ the subject.
4. Replace $y$ with $f^{- 1} (x)$ to rewrite as $f^{- 1} (x) = . . .$ .
If you knew that $f (3) = 10$ , how would you find $f^{- 1} (10)$ ?
An inverse function 'undoes' the operations of the original function.
So if $f$ turns 3 into 10, then $f^{- 1}$ turns 10 back into 3.
I.e., if $f (3) = 10$ , then $f^{- 1} (10) = 3$ .
What is ${ff}^{- 1} (x)$ equal to?
${ff}^{- 1} (x)$ is a composite function of $f$ and its inverse $f^{- 1}$ .
An inverse function 'undoes' the operations of the original function (and vice versa).
So ${ff}^{- 1} (x) = x$ .
(Note that $f^{- 1} f (x) = x$ as well.)
True or false?
Completing the square can help find inverse functions.
True.
Completing the square can help rearrange a quadratic expression in order to find its inverse function.

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