2-6-2 Function Notation Explained
Key Concepts of Function Notation
Function notation is a way to represent functions using symbols. The key concepts include:
- Function Definition: A function is a relation where each input has a unique output.
- Function Notation: Representing a function using symbols like \( f(x) \).
- Evaluating Functions: Finding the output of a function for a given input.
- Domain and Range: The set of all possible inputs (domain) and outputs (range) of a function.
1. Function Definition
A function is a special type of relation where each input (independent variable) has exactly one output (dependent variable). For example, the function \( f(x) = x^2 \) means that for each input \( x \), the output is \( x^2 \).
Example:
If \( f(x) = x^2 \), then \( f(3) = 3^2 = 9 \).
2. Function Notation
Function notation, such as \( f(x) \), is used to represent a function. The letter \( f \) is the name of the function, and \( x \) is the input variable. The notation \( f(x) \) represents the output of the function when the input is \( x \).
Example:
If \( f(x) = 2x + 3 \), then \( f(4) = 2(4) + 3 = 8 + 3 = 11 \).
3. Evaluating Functions
Evaluating a function means finding the output for a given input. This involves substituting the input value into the function and performing the necessary calculations.
Example:
Evaluate \( g(x) = 3x - 5 \) for \( x = 2 \):
\[ g(2) = 3(2) - 5 = 6 - 5 = 1 \]
4. Domain and Range
The domain of a function is the set of all possible input values, and the range is the set of all possible output values. For example, the domain of \( f(x) = \sqrt{x} \) is \( x \geq 0 \), and the range is \( f(x) \geq 0 \).
Example:
Find the domain and range of \( h(x) = \frac{1}{x} \):
The domain is all real numbers except \( x = 0 \) (since division by zero is undefined).
The range is all real numbers except \( y = 0 \) (since \( \frac{1}{x} \) can never be zero).
Examples and Analogies
To better understand function notation, consider the following analogy:
Imagine a function as a machine that takes an input, processes it, and produces an output. Function notation is like labeling the machine and its input/output ports. For example, \( f(x) = x^2 \) is like a machine that squares any number you put into it.
Practical Applications
Function notation is used in various real-world applications, such as:
- Modeling physical phenomena in physics.
- Analyzing data in statistics.
- Programming in computer science.
Example:
In physics, the distance \( d \) traveled by an object can be modeled by the function \( d(t) = \frac{1}{2}at^2 \), where \( a \) is acceleration and \( t \) is time. Evaluating \( d(t) \) at different times helps in predicting the object's position.