Question

What is a function?

Short answer

Answer: A function in mathematics is a rule that assigns each element in one set (domain) to exactly one element in another set (codomain). Functions are usually represented as f(x), where x is the input, and f(x) represents the output value. An example of a function is f(x) = 2x + 3. To determine if a relationship is a function, it must satisfy two main rules: (1) each input has exactly one output, and (2) no two distinct inputs share the same output. The Vertical Line Test can be used to verify if a given graph represents a function - every vertical line crossing the graph of the relationship must intercept the graph at most once.

Step-by-step solution

Definition of a function

A function is a rule that assigns each element in one set, called the domain, to exactly one element in another set, called the codomain. The set of all possible outputs is called the range of the function.

Function notation

Functions are usually represented as f(x), where x is the input (from the domain), and f(x) represents the output value (from the range). The letter 'f' can be any other letter as well.

Example of a function

Let us consider the function f(x) = 2x + 3. This means that for any input value x from the domain, the output value will be twice the value of x plus three. To find the output for a specific input, substitute the value of x into the expression and simplify. For example: Input (x) | Output f(x) ------------|--------------- 1 | f(1) = 2(1) + 3 = 5 2 | f(2) = 2(2) + 3 = 7 3 | f(3) = 2(3) + 3 = 9

Function rules

Keep in mind that there are two main rules for a relation to be considered a function: 1. Each input has exactly one output. 2. No two distinct inputs share the same output. If these rules are not satisfied, the relationship is not a function.

Vertical Line Test

A relationship between sets A and B in the Cartesian plane can also be considered a function if every vertical line crossing the graph of the relationship intercepts the graph at most once. This is a useful test to verify if a given graph represents a function or not.

Key concepts

Definition of a Function

A function is a fundamental concept in mathematics that describes a special kind of relationship between sets. Imagine a function as a reliable machine where you insert an input and it gives you an output. But here's the catch: each input is assigned to one and only one output. To be more technical, it is a rule that takes every element (we call this the domain) and matches it to a unique element in another set (known as the codomain). The set of outputs that we get from using the function is called the range.

Understanding functions is crucial because they express the idea that one quantity completely depends on another. For instance, consider temperature as a function of time throughout the day. At each moment in time, there's a specific temperature. Similarly, in mathematics, for every input value, there is a corresponding output value when a function is applied.

Function Notation

When you come across expressions like f(x), you're looking at function notation. It's the way we write down functions to show the relationship between the input x and the function f which generates the output. This notation tells you that the role of f is to take x, apply a certain rule, and produce an output, f(x). The cool thing is you can replace f with any letter, such as g or h, and the meaning remains the same.

For instance, if f(x) = 2x + 3, we understand that the function f will double the input x and then add three to give us the output. It's an elegant way of showing what happens to x when it goes through our function machine. You can think of it like a mathematical recipe; you follow certain steps with your ingredient x to end up with your mathematical meal f(x).

Vertical Line Test

The vertical line test is a handy visual tool to determine whether a curve on a graph represents a function. What you do is draw vertical lines across the graph. If each vertical line hits the curve only once, ta-da, you've got a function! This ensures that for each input (which corresponds to the x-coordinate), there is only one output (the y-coordinate on the curve).

If a vertical line intersects the graph more than once, it means there are multiple outputs for a single input, breaking our function rule. So, the vertical line test is a quick and easy way to see if we're dealing with a function or not. It's like swiping a comb through the hair of a graph's curve; each tooth (or line) should only make contact once if the hair (the curve) represents a function.