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 characteristic of a function is that for every input value, there is only one corresponding output value. For example, in the function f of x equals x squared, when we input negative 2, we get exactly one output, which is 4. This one-to-one mapping from input to output is what defines a function.
Functions can be represented in various ways. The most common representations include equations, such as y equals 2x plus 1, graphs that visually show the relationship between inputs and outputs, tables that list pairs of input and output values, mapping diagrams, and arrow diagrams. In mathematics, we commonly use the notation f of x to represent the output of a function f for an input x. For example, in the function f of x equals 2x plus 1, when x equals 0, f of 0 equals 1, and when x equals 1, f of 1 equals 3, as shown in our table.
Let's understand what makes a relation a function. A relation is a function if and only if each input value corresponds to exactly one output value. 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. For example, y equals x squared is a function because any vertical line intersects the graph at exactly one point. Similarly, y equals absolute value of x is also a function. However, the circle with equation x squared plus y squared equals 1 is not a function because a vertical line can intersect it at two points, as shown in our example. Likewise, y squared equals x is not a function because some input values have two corresponding outputs.
Two important concepts in functions are domain and range. The domain of a function 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 some examples. For the function f of x equals x squared, the domain includes all real numbers since we can square any number. The range, however, is limited to non-negative numbers, from zero to infinity, because a square is always non-negative. For the function g of x equals 1 divided by x, the domain includes all real numbers except zero, since division by zero is undefined. Similarly, the range includes all real numbers except zero. For the square root function, h of x equals the square root of x, the domain is restricted to non-negative numbers, and the range is also non-negative numbers.
To summarize what we've learned about functions: A function is a special type of relation where each input value corresponds to exactly one output value. Functions can be represented in various ways, including equations, graphs, tables, and mapping diagrams. We can use the vertical line test to determine if a graph represents a function. The domain of a function is the set of all valid input values, while the range is the set of all possible output values. Functions are fundamental mathematical concepts with applications in science, engineering, economics, and many other fields. They help us model relationships between quantities and make predictions based on those relationships.