🟣 Functions and Their Properties
Master the concept of functions! Learn to identify, represent, and analyze functions, domains, and ranges with precision.
📐 Understanding Functions
A function is a special type of mathematical rule that assigns exactly one output for each input.
Key Concept
- If you input a number into a function, you get only one result.
- Every function can be written as $f(x)$, where:
- $x$ is the input (or independent variable).
- $f(x)$ is the output (or dependent variable).
Example: Consider the function:
$$f(x) = 2x + 3$$
- If $x = 1$, then $f(1) = 2(1) + 3 = 5$.
- If $x = -2$, then $f(-2) = 2(-2) + 3 = -1$.
A function ensures that for every $x$, there is only one $f(x)$ (no multiple outputs for a single input).
Not a function: $y^2 = x$ (because one $x$-value can have two $y$-values).
📊 Ways to Represent a Function
A function can be represented in multiple ways:
- Equation Form: Example: $f(x) = 3x - 5$.
- Table Form: Lists input-output pairs.
$x$ $f(x)$ 0 -5 1 -2 2 1 - Graph Form: A visual representation by plotting points.
- Verbal Description: For example, "A function that doubles the input and adds 3."
📏 The Vertical Line Test
A function's graph must pass the vertical line test:
- If a vertical line touches the graph at only one point everywhere, it is a function.
- If a vertical line crosses the graph more than once, it is not a function.
Examples:
- $y = x^2$ ✅ Function (each $x$ gives one $y$).
- $x = y^2$ ❌ Not a function (some $x$-values yield two $y$-values).
🔄 Types of Functions
1️⃣ Linear Functions
A straight-line function in the form:
$$f(x) = mx + b$$
$m$ is the slope and $b$ is the y-intercept.
Example: $f(x) = 2x + 3$ (slope = 2, y-intercept = 3).
2️⃣ Quadratic Functions
Forms a parabola:
$$f(x) = ax^2 + bx + c$$
Example: $f(x) = x^2 - 4x + 3$ (a U-shaped curve with a vertex and axis of symmetry).
3️⃣ Absolute Value Functions
Form a V-shape:
$$f(x) = |x|$$
Example: $f(2) = 2$, $f(-3) = 3$.
4️⃣ Piecewise Functions
Defined by different rules for different intervals of $x$:
Example:
$$f(x)= \begin{cases} x+2, & x < 0 \\ x^2, & x \ge 0 \end{cases}$$
For $x = -1$, $f(-1)=1$; for $x = 2$, $f(2)=4$.
🎯 Domain and Range
Domain (Input Values)
The set of all possible $x$-values that can be used as input.
Example: For $f(x)=\sqrt{x}$, the domain is $x\ge 0$.
Range (Output Values)
The set of all possible $f(x)$-values (outputs).
Example: For $f(x)=x^2$, the range is $y\ge 0$.
🎯 Practice Questions
Test your understanding!
🔥 Challenge Questions
Push your limits!