Functions

Warm up:

Function Machines
(Care of www.mathsgenie.co.uk)

Function Machines Video
Practice Questions
Textbook Exercise
(Care of corbettmaths.com)

Describing functions:

A function is a mathematical description, rule or law, between variables. Functions take an input value, i.e., $x$ and converts it into an output value $y$.

Examples:

$y=3x$;

$f\left(x\right)=x^2$;

$I=Anqv$

Note, the notation $f\left(x\right)$ is read "f of x".

Example 1:

If $f\left(x\right)=2x+7$, then find $f\left(2\right)$, $f\left(-1\right)$, and $f\left(\mathrm{⅗}\right)$.

Solution:

$f\left(2\right)=2\times 2+7=11$

$f\left(-1\right)=2\times -1+7=5$

$f\left(\mathrm{⅗}\right)=2\times \mathrm{⅗}+7=\frac{41}{5}=8\mathrm{⅕}$

Example 2:

If $f\left(x\right)=x^2-9$, then find $f\left(x\right)=-9$, $f\left(x\right)=16$, and $f\left(x\right)=-10$.

Solution:

Example 3:

If h(x) = x² - 3x, then find h(x) = 10, h(x) = -2, h(x) = 0.

Solution:

h(x) = 10
x² - 3x = 10
x² - 3x - 10 = 0
(x - 5)(x + 2) = 0
x = {-2, 5}

h(x) = -2
x² - 3x = -2
x² - 3x + 2 = 0
(x - 1)(x - 2) = 0
x = {1, 2}

h(x) = 0
x² - 3x = 0
x(x - 3) = 0
x = {0, 3}

Example 4:

If g(x) = tan x, then find g(x) = 1.

Solution:

g(x) = 1
tan x = 1
x = arctan 1
x = pi/4 + pim where $m\in \mathrm{\mathbb{Z}}$.

Questions:

Care of mmerevise.co.uk

Further Materials

Inverse Functions

To find the inverse of a function, a graphical or algebraic method can be used. Graphically the inverse function can be found by reflecting the function in $y=x$. Note: the domain and range are swopped with either method.

To find the inverse function, $f^{-1}\left(x\right)$ algebraically we follow this procedure:

1. replace $f\left(x\right)$ with $y$;
2. rearrange this equation to make $x$ the subject;
3. then replace $x$ with $f^{-1}\left(x\right)$ and $y$ with $x$.

Examples:

from corbettmaths.com:

Inverse Functions video

Practice Questions

Textbook Exercise

Further Reading:

from mathsisfun.com

from math.net

from wikipedia.org

Notes:

The substitution of $f\left(x\right)$ with $y$ is just to make it easier to rearrange in step 2.

Function & Graph Transformations

Modulus Function

$f\left(x\right)=\left|x\right|$

Composite Functions

Prerequisite Material:

Functions

Composite Functions:

Definition: a function made of up of two or more other functions, where the output of one is the input to the other and so on.

Video
from corbettmaths.com

Example:
If $f\left(x\right)=x+2$, then $f\left(3\right)=3+2=5$.
We could take this output and enter it into another function, say, $g\left(x\right)=3x-4$.
Entering the value from $f\left(x\right)$, then $g\left(5\right)=3\times 5-4=11$.

The same result can be found by substituting $f\left(x\right)$ into $g\left(x\right)$

$g\left(f\left(x\right)\right)=g(x+2)=3(x+2)-4=3x+2$

So, if $x=3$ then $g\left(f\left(3\right)\right)=3\times 3+2=11$.
That is the same as before.

An alternate way of writing $g\left(f\left(x\right)\right)$ can be written as $gf\left(x\right)$.
Note: the function $f$ is worked out before $g$.

In general $gf\left(x\right)$ is not the same as $fg\left(x\right)$.

Inverse functions have the following property:

Questions:

Practice Questions from corbettmaths.com
Textbook Questions from corbettmaths.com
GCSE Questions & Answers from piacademy.co.uk
Further Maths Video from corbettmaths.com
Further Maths Practice Questions from corbettmaths.com

Further Reading:

Inverse Functions

Domain and Range

Domain for the above is $\{x:-3<x\le 1\}$,

and the range is $\left\{f\right(x):-4\le f(x)<0\}$.

Domains and Ranges Video

Practice Questions [ MS ]