What is a function? A function is a relation between a set of inputs and a set of possible outputs where each input is related to exactly one output. The key characteristic of a function is that each input maps to exactly one output. In this diagram, we can see inputs 1, 2, 3, and 4 mapping to outputs A, B, and C. Notice that different inputs can map to the same output, as shown by inputs 2 and 4 both mapping to output B. This is allowed in a function. However, a single input cannot map to multiple outputs.
Let's look at function notation and examples. A function is typically written as f from X to Y, where X is the domain or input set, and Y is the codomain or output set. We often write y equals f of x, where x is the input and y is the output. Common examples of functions include linear functions like f of x equals 2x plus 3, quadratic functions like g of x equals x squared, and trigonometric functions like h of x equals sine of x. On the right, we can see the graph of the linear function f of x equals 2x plus 3. For any input x, we get exactly one output y. As we move along the x-axis, the function gives us exactly one y-value for each x-value. This is the fundamental property of a function - each input maps to exactly one output.
Let's distinguish between what is a function and what is not a function. A function requires that each input maps to exactly one output. If any input maps to multiple outputs, then the relation is not a function. We can use the vertical line test to determine if a graph represents a function. If any vertical line intersects the graph more than once, then it's not a function. Looking at our examples, the parabola on the left passes the vertical line test - each vertical line intersects it exactly once. This means each x-value has exactly one y-value, confirming it's a function. The circle on the right fails the vertical line test - a vertical line can intersect it twice. This means some x-values have two different y-values, so it's not a function. Remember, the key characteristic of a function is that each input must have exactly one output.
Let's explore the domain and range of functions. The domain is the set of all possible input values, or x-values, for which the function is defined. The range is the set of all possible output values, or y-values, that can be produced by the function. Let's look at the function f of x equals the square root of x as an example. For this function, the domain is restricted to x greater than or equal to zero, because we cannot take the square root of negative numbers. The range is also restricted to y greater than or equal to zero, because the square root operation always gives a non-negative result. On the graph, we can see that the function is only defined for x-values to the right of the y-axis, which represents the domain. And the function only produces y-values above or on the x-axis, which represents the range. As we trace along the function, each input in the domain maps to exactly one output in the range, following our definition of a function.
To summarize what we've learned about functions: A function is a special type of relation where each input maps to exactly one output. We use function notation like f of x to represent the output when x is the input. The vertical line test helps us determine if a graph represents a function - if any vertical line intersects the graph more than once, it's not a function. Every function has a domain, which is the set of valid input values, and a range, which is the set of possible output values. Functions are fundamental mathematical tools used extensively in science, engineering, economics, and countless real-world applications. They help us model relationships between quantities and make predictions based on those relationships.