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 the value 1, we get exactly one output, which is also 1. This one-to-one mapping from input to output is what defines a function.
When discussing functions, we need to understand two important concepts: 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 result from the function. For our example function f of x equals x squared, the domain includes all real numbers since we can square any real number. The range, however, only includes non-negative real numbers, because squaring any number always gives a result that is zero or positive. On the graph, the domain is represented by the x-axis, and the range is represented by the y-values that the function can produce.
Let's distinguish between functions and non-functions. A function must have exactly one output for each input. 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. On the left, we have a parabola which is a function - any vertical line crosses it exactly once. On the right, we have a circle which is not a function - a vertical line can intersect it at two points, meaning that a single input value would have two different outputs. This violates the definition of a function.
There are many different types of functions, each with unique properties. 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 follow the form f of x equals a x squared plus b x plus c, and they create parabolas. Exponential functions have the form f of x equals a times b to the power of x, where b is the base. These grow or decay at a rate proportional to their current value. On our graph, the blue line represents a linear function, the red curve is a quadratic function, and the green curve shows an exponential function. Each type of function has specific applications in mathematics, science, and real-world problem-solving.
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. The domain of a function is the set of all possible input values, while the range is the set of all possible output values. 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, it's not a function. There are many types of functions, including linear, quadratic, and exponential functions, each with unique properties and applications. Functions are powerful mathematical tools that help us model relationships and solve problems in various fields, from physics and engineering to economics and data science.