Functions and Their Properties
1️⃣ Understanding Functions
A function is a special type of mathematical rule that assigns exactly one output for each input.
- 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).
2️⃣ 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.”
3️⃣ 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.
Example:
- \( y = x^2 \) ✅ Function (each \( x \) gives one \( y \)).
- \( x = y^2 \) ❌ Not a function (some \( x \)-values yield two \( y \)-values).
4️⃣ 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 \( x = 2 \), \( f(2)=4 \).
5️⃣ 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 – Fundamental
- Determine whether the following are functions: \( y = x^3+2x-1 \), \( x = y^2 \), \( y=|x| \).
- Find the domain and range of: \( f(x)=2x-3 \) and \( g(x)=\sqrt{x+4} \).
- Which of the following pass the vertical line test?
- Identify whether the following functions are linear, quadratic, or absolute value: \( y=4x-7 \), \( y=x^2-6x+5 \), \( y=|x-2| \).
- Find \( f(3) \) if \( f(x)=5x-2 \).
- If \( f(x)=x^2-2x+1 \), find \( f(-2) \).
- Graph the function \( f(x)=|x-3| \).
- Identify the slope and y-intercept of \( f(x)=-3x+5 \).
- Evaluate \( g(x)=3x^2-2x+4 \) for \( x=-1 \).
- Write a piecewise function that represents: \( f(x)=x+2 \) for \( x<0 \) and \( f(x)=x^2 \) for \( x\ge0 \).
Challenging Problems – Take It Up a Notch!
- Given \( f(x)=\frac{3x+2}{x-1} \), determine its domain.
- Find all \( x \) such that \( f(x)=g(x) \) for \( f(x)=2x+1 \) and \( g(x)=x^2-3 \).
- A company’s revenue is \( R(x)=500x-x^2 \). Find the maximum revenue.
- Find the equation of a function that passes through \( (2,5) \) and \( (4,9) \) with a slope of 2.
- The population of a city follows \( P(t)=5000+200t \) (where \( t \) is years since 2000). Find the population in 2030.
7️⃣ Summary
- ✅ A function gives one output for each input.
- ✅ The vertical line test determines whether a graph represents a function.
- ✅ Functions can be represented as equations, tables, graphs, or verbally.
- ✅ Domain and range define the possible inputs and outputs of a function.
8️⃣ Great Work!
Fantastic work! You're ready for the next challenge: More Advanced Function Topics!