Update Unit 6: Functions.md

Grade 10/Math/MPM2DZ/Unit 6: Functions.md

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+# 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:
+
+```math
+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.
+
+
+
+