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 every input value corresponds to precisely one output value. In this example, we have a function f of x equals zero point five x squared. When we input the value 2, the function gives us exactly one output, which is 2. This one-to-one mapping from input to output is what defines a function.
Functions can be represented in multiple ways. The most common representations include equations, tables of values, graphs, and mappings. Here we see a mapping diagram for the function f of x equals x squared. Each input value on the left is mapped to exactly one output value on the right. For example, when x equals 1, the output is 1. When x equals 2, the output is 4. When x equals 3, the output is 9. And when x equals 4, the output is 16. This mapping clearly shows how each input corresponds to exactly one output, which is the defining characteristic of a function.
To determine if a relation is a function, we can use the vertical line test. 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 is a function. When we draw a vertical line at any x-value, it intersects the parabola at exactly one point. On the right, we have a circle which is not a function. When we draw a vertical line, it intersects the circle at two points, meaning that for a single input value, we get two different outputs. This violates the definition of a function, which requires that each input corresponds to exactly one output.
Two important concepts associated with functions are domain and range. 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 consider the function f of x equals the square root of x. For this function, the domain is restricted to non-negative numbers, because we cannot take the square root of a negative number. So the domain is x greater than or equal to zero, shown by the red line on the x-axis. The range is also restricted to non-negative numbers, because the square root of a non-negative number is always non-negative. So the range is y greater than or equal to zero, shown by the green line on the y-axis. The red shaded region represents values that are not in the domain of this function.
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 multiple ways, including equations, tables of values, graphs, and mappings. The vertical line test is a useful tool to determine if a graph represents a function - if any vertical line intersects the graph at more than one point, then it's not a function. Two important concepts associated with functions are domain and range. The domain is the set of all valid input values, while the range is the set of all possible output values. Understanding these fundamental concepts is essential for working with functions in mathematics and real-world applications.