5.2 KiB
Unit 6: Functions
Functions
Function: a correspondnce between two sets of elements.
It “links” each element between the first set with one and only
one element in the second set. - The first set is called the
Domain (values of x) - The second set is
called the Range (values of y)
A relation is something that relates a variable to another. It is slightly different than a function. All functions are relations, but not all relations are functions!
Vertical Line Test: A test to check whether or not a
relation is a function. If a vertical line parallel to
the y-axis can be drawn through any 2 points on a relation, then that
relation is not a function.
Horizontal Line test: A test to check whetehr or not the
inverse of a function is also a function. If a horizontal
line parallel to the x-axis can be drawn through any 2 points
on a function, then the inverse of that function is not a
function.
Types Of Functions
| Function | Base Form | Domain | Range |
|---|---|---|---|
| Linear Function | \(`y=x`\) | \(`\{ x \mid x \in \mathbb{R} \}`\) | \(`\{y \mid y \in \mathbb{R} \}`\) |
| Quadratic Function | \(`y=x^2`\) | \(`\{x \mid x \in \mathbb{R} \}`\) | \(`\{y \mid y \ge 0, y \in \mathbb{R} \}`\) |
| Cubic Function | \(`y=x^3`\) | \(`\{x \mid x \in \mathbb{R} \}`\) | \(`\{y \mid y \in \mathbb{R} \}`\) |
| Square Root Function | \(`y= \sqrt{x}`\) | \(`\{x \mid x \ge 0, x \in \mathbb{R}\}`\) | \(`\{y \mid y \ge 0, y \in \mathbb{R}\}`\) |
| Absolute Value Function | \(`y = \| x \|`\) | \(`\{ x \mid x \in \mathbb{R}\}`\) | \(`\{y \mid y \ge 0, y \in \mathbb{R}\}`\) |
| Reciprocal Function | \(`y = \dfrac{1}{x}`\) | \(`\{x \mid x =\not 0, x \in \mathbb{R}\}`\) | \(`\{y \mid y =\not 0, y \in \mathbb{R}\}`\) |
| Exponential Function | \(`y = 2^x`\) | \(`\{x \mid x \in \mathbb{R}\}`\) | \(`\{y \mid y \gt 0, y \in \mathbb{R}\}`\) |
| Logarithmic Function (Inverse of the Exponential Function) | \(`y = \log (x)`\) | \(`\{x \mid x \gt 0, x \in \mathbb{R}\}`\) | \(`\{y \mid y \in \mathbb{R}\}`\) |
To get base points, simply choose integers in the range \(`-3 \le x \le 3`\) (cause they are small, and \(`0`\) is always a great choice), and plug it in as the x-value and you will get the value from the specific function, and that will be one of your base points.
Piecewise Functions
These are functions that are made of multiple functions. Thus they have a specified domain for each piece. Examples include:
f(x) = \begin{cases}
x^2 & \text{if } x \lt 2 \\
6 & \text{if } x = 2 \\
10 - x & \text{if } 2 \lt x \le 6 \\
\end{cases}
\quad \text{and} \quad
f(x) = \begin{cases}
-4 & \text{if } -8 \lt x \le -3 \\
x - 2 & \text{if } -3 \lt x \lt 0 \\
\sqrt{x} & \text{if } 0 \lt x \le 8 \\
\end{cases}Reminder, if the function does not include a point, make sure to use a empty dot (\(`\circ`\)) instead of a filled dot (\(`\bullet`\)).
Functions With Other Functions (Compositions of Functions)
Let \(`f(x), g(x)`\) represent two functions. We can do other things with functions, such as: - Add two functions: \(`f(x) + g(x)`\) - Subtract two functions: \(`f(x) - g(x)`\) - Multiply two functions: \(`f(x) \times g(x)`\) - Divide two functions: \(`\dfrac{f(x)}{g(x)}`\) - Find the inverse of a function: \(`f^{-1}(x)`\)
It is also possbile to put functions within of functions, such as \(`f(g(x))`\). This can also be represented as \(`(f \circ g)(x)`\). \(`f(g(x))`\) is an example of a Composite Function.
Transformation Of Functions.
Any functions can be repsented in a form \(`\Huge y = af[k(x-d)] + c`\)
The mapping rule states for a base point \(`(x, y)`\), the new point will be \(`(\dfrac{1}{k}x + d, ay + c)`\).
Vertical Translations
If \(`d \lt 0`\), the graph moves to the left horizontally \(`d`\) units.
If \(`d \gt 0`\), the graph moves to the right horizontally \(`d`\) units.
Horizontal Translations
If \(`c \lt 0`\), the graph moves down vertically \(`d`\) units.
if \(`c \gt 0`\), the graph moves up vertically \(`d`\) units.
Vertical Stretch/Compressions
If \(`0 \lt a \lt 1`\), then the graph gets compressed by a factor of \(`a`\).
If \(`a \gt 1`\), then the graph gets stretched by a factor of \(`a`\).
Horizontal Stretch/Compressions
If \(`0 \lt k \lt 1`\), then the graph gets stretched by a factor of \(`\dfrac{1}{k}`\)
If \(`k \gt 1`\), then the graph gets compressed by a factor of \(`\dfrac{1}{k}`\).
Vertical/Horiztonal Relfections
If \(`a \lt 0`\), then the graph gets reflected over the x-axis.
If \(`k \lt 0`\), then the graph gets reflected over the y-axis.
Finding Inverse
To find the inverse, simply replace \(`f(x)`\) with \(`x`\) and \(`x`\) with \(`y`\). Then try to model the equation for \(`y`\). That will be the inverse function of the original function.
After swtiching them, here a few tips for some of the functions.
For the quadratic function, complete the
square and model the equation for \(`y`\).
For the reciprocal function, cross multiply, then factor
out the \(`y`\), and then model the
equation for \(`y`\). State
restrictions whenever necessary.
For the square root funtion, make sure to state your
restrictions since without, it becomes the inverse of a quadratic
equation.
Remember to change to \(`f^{-1}`\) only if the relation is a function for the inverse.