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 the domain of a function .

The domain of a function is the set of all inputs that are allowed to be put into the function.

Define the range of a function .

The range of a function is the set of all outputs that the function gives out, based on the set of all inputs (the range) that can be put in.

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.

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Functions (Cambridge (CIE) IGCSE 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 the domain of a function.
The domain of a function is the set of all inputs that are allowed to be put into the function.
True or False?
The domain of $f (x) = \frac{1}{x}$ is "all values of $x$ ".
False.
$f (x) = \frac{1}{x}$ is not defined when $x = 0$ .
Therefore zero must be excluded from the domain of $f (x)$ .
$f (x) = \frac{1}{x}$ , $x \neq 0$ .
Define the range of a function.
The range of a function is the set of all outputs that the function gives out, based on the set of all inputs (the range) that can be put in.
True or false?
The function $f (x) = x^{2}$ with domain "all values of $x$ " also has a range which is "all values of $x$ ".
False.
Whether $x$ is positive or negative, $f (x) = x^{2}$ can only ever be positive or zero.
The domain is 'all values of $x$ ' and the range is $f (x) \geq 0$ .
Out of the domain and range of a function, which one is expressed in terms of $x$ , and which one is expressed in terms of $y$ or $f (x)$ ?
The domain is expressed in terms of $x$ (e.g. $x \neq 0$ ).
The range is expressed in terms of $y$ or $f (x)$ (e.g. $y \geq 0$ or $f (x) \geq 0$ ).
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.)
What is the domain of $f^{- 1} (x)$ ?
The domain of $f^{- 1} (x)$ has the same values as the range of $f (x)$ .
When writing down this domain, you must write it in terms of $x$ .
What is the range of $f^{- 1} (x)$ ?
The range of $f^{- 1} (x)$ has the same values as the domain of $f (x)$ .
When writing down the range of $f^{- 1} (x)$ , use $f^{- 1} (x)$ as the variable, not $x$ .
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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