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 exactly one corresponding output value. For example, in the function f of x equals x squared, when we input negative 1, we get an output of 1. Similarly, when we input positive 1, we also get an output of 1. This demonstrates that different inputs can map to the same output, but each input must map to exactly one output.
When we represent a function, we typically use the notation f of x equals some expression. For example, f of x equals x squared. 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. For our function f of x equals x squared, the domain includes all real numbers. The range is the set of all possible output values that the function can produce. For f of x equals x squared, the range includes all non-negative real numbers, since a square is always greater than or equal to zero. When we input 1.5 into our function, we get an output of 2.25, which is 1.5 squared.
There are many different types of functions, each with their own characteristics. Linear functions have the form f of x equals m x plus b, where m is the slope and b is the y-intercept. They create straight lines when graphed. Quadratic functions have the form f of x equals a x squared plus b x plus c, where a, b, and c are constants. They create parabolas when graphed. Exponential functions have the form f of x equals a raised to the power of x, where a is a positive constant. They grow or decay at a rate proportional to their current value. Each type of function has unique properties and applications in various fields of mathematics and science.
To determine if a relation is a function, we can use the vertical line test. If any vertical line intersects a graph at more than one point, then the graph does not represent a function. For example, a parabola passes the vertical line test because each vertical line intersects it at most once. However, a circle fails the test because some vertical lines intersect it twice. Another important property is whether a function is one-to-one. A one-to-one function has each output corresponding to exactly one input. We can use the horizontal line test to check this: if any horizontal line intersects the graph more than once, the function is not one-to-one. For instance, a cubic function is one-to-one, but a quadratic function is not because different inputs can produce the same output.
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. We typically represent functions using the notation f of x equals some expression. Every function has a domain, which is the set of all possible input values, and a range, which is the set of all possible output values. There are many types of functions, including linear, quadratic, and exponential functions, each with their own unique properties. We can use the vertical line test to determine if a graph represents a function, and the horizontal line test to check if a function is one-to-one. Functions are fundamental mathematical tools used in various fields to model relationships between quantities.