A function is a relation between a set of inputs and a set of outputs where each input is related to exactly one output. The key rule of a function is that each input must have exactly one corresponding output. In this diagram, we can see inputs on the left mapped to outputs on the right. Notice that different inputs can map to the same output, as shown by inputs 2 and 3 both mapping to output B. This is allowed in a function. What's not allowed is having one input map to multiple outputs.
Let's look at function notation and some examples. We typically write a function as f of x equals y, which means the function f takes an input x and produces an output y. Some common examples of functions include f of x equals x squared, g of x equals 2x plus 3, and h of x equals sine of x. Here we're visualizing the parabola function f of x equals x squared. For each input value of x, the function produces exactly one output value y. As we move the point along the curve, you can see that different x values produce different y values according to the function rule. This is the essence of a function - a rule that assigns exactly one output to each input.
Let's distinguish between functions and non-functions. A function must assign exactly one output to each input. If any input has more than one possible output, 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 at more than one point, then the graph does not represent a function. On the left, we have a parabola which passes the vertical line test - any vertical line will intersect it at most once. This is a function. On the right, we have a circle which fails the vertical line test - a vertical line can intersect it at two points, as shown by the green line. This means some x-values have two different y-values, so a circle is not a function.
Now let's discuss the domain and range of functions. The domain is the set of all possible input values for which the function is defined. The range is the set of all possible output 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 y greater than or equal to zero, because the square root operation always gives non-negative results. As we move the point along the function, you can see that both x and y remain non-negative. Understanding domain and range is crucial for properly applying functions in various contexts.
To summarize what we've learned about functions: A function is a special type of relation where each input has exactly one corresponding output. We can represent functions using notation like f of x equals y, and visualize them as graphs. The vertical line test helps us identify whether a graph represents 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 that help us model relationships in science, engineering, economics, and countless other fields.