Math Lesson 6: Functions
6.1 What is a function?
A function is a special relationship between two sets of numbers, called the domain and range. For each input value in the domain, there is exactly one output value in the range.
Think of a function like a machine. You put an input into the machine, and it produces an output based on a specific set of rules.
Example:
Let's say we have a function that doubles the input number. The domain is all real numbers, and the range is also all real numbers. If we input 5, the output is 10 (5 * 2 = 10). If we input -3, the output is -6 (-3 * 2 = -6).
6.2 Function notation
We use a special notation to represent functions. The most common notation is:
f(x) = ...
where:
- f is the name of the function
- x is the input variable
- ... represents the rule that defines the function
Example:
The function that doubles the input can be written as: f(x) = 2x.
To find the output for a specific input, we substitute the input value for x. For example, f(3) = 2 * 3 = 6.
4.3 The basic functions, formulas and graphs
There are several basic functions that are essential to understand in algebra. Here are some of the most common ones:
Linear functions
A linear function is a function that can be represented by a straight line when graphed. The general form of a linear function is:
f(x) = mx + c
where:
- m is the slope of the line (the rate of change)
- c is the y-intercept (the point where the line crosses the y-axis)
Example:
f(x) = 3x + 2 is a linear function with a slope of 3 and a y-intercept of 2.
Exercise: Linear Function
Find the value of f(4) for the linear function: f(x) = 2x - 1
Quadratic functions
A quadratic function is a function that can be represented by a parabola when graphed. The general form of a quadratic function is:
f(x) = ax2 + bx + c
where:
- a, b, and c are constants
Example:
f(x) = x2 - 4x + 3 is a quadratic function.
Exercise: Quadratic Function
Find the value of f(2) for the quadratic function: f(x) = x2 + 3x - 2
Exponential functions
An exponential function is a function where the input variable appears in the exponent. The general form of an exponential function is:
f(x) = ax
where:
- a is a constant called the base (a > 0 and a ≠ 1)
Example:
f(x) = 2x is an exponential function with a base of 2.
Exercise: Exponential Function
Find the value of f(3) for the exponential function: f(x) = 3x
4.4 Inverse functions
An inverse function "undoes" the original function. If we apply a function and then its inverse, we get back to the original input.
Notation: The inverse of a function f(x) is denoted as f-1(x).
Example:
Let's say f(x) = 2x + 1. The inverse function is f-1(x) = (x - 1) / 2.
If we input 3 into f(x), we get f(3) = 2 * 3 + 1 = 7.
If we then input 7 into f-1(x), we get f-1(7) = (7 - 1) / 2 = 3, which is our original input.
4.5 The logarithmic function
The logarithmic function is the inverse of the exponential function. It helps us find the exponent to which a base must be raised to get a certain number.
Notation: loga(x) = y
This means that ay = x.
Example:
log2(8) = 3, because 23 = 8.
Next lesson