What is a function? A function is a relation that assigns each input exactly one output. The set of all possible inputs is called the domain, while the set of all possible outputs is called the codomain. The range is the set of actual outputs that are used. The key rule of a function is that each input must map to exactly one output. In this diagram, we can see how inputs from the domain are mapped to outputs in the codomain through the function f.
Let's look at some examples of functions. Linear functions like f(x) equals 2x plus 3, quadratic functions like f(x) equals x squared, and trigonometric functions like f(x) equals sine of x are all valid functions because each input x has exactly one output. However, not all relations are functions. For example, a circle represented by the equation x squared plus y squared equals 1 is not a function because for most x-values, there are two possible y-values. Similarly, y squared equals x is not a function because each positive x maps to both a positive and negative y. 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. As you can see, the vertical line intersects the circle twice, confirming it's not a function.
Let's explore function notation and evaluation. We write a function using the notation f of x equals some expression, like x squared plus 3x minus 2. To evaluate a function at a specific input value, we substitute that value for x in the function expression. For example, to find f of 2, we substitute 2 for x in our function. So f of 2 equals 2 squared plus 3 times 2 minus 2, which equals 4 plus 6 minus 2, giving us 8. The domain of a function is the set of all possible input values. For this quadratic function, the domain is all real numbers. The range is the set of all possible output values. For this parabola, the range is y greater than or equal to negative 11 over 4, which occurs at the minimum point when x equals negative 3 over 2. Watch as we move the input value along the x-axis and see how the function value changes.
Let's explore different types of functions. Linear functions have the form f(x) equals mx plus b, where m is the slope and b is the y-intercept. Quadratic functions follow the form f(x) equals ax squared plus bx plus c, creating parabolas. Cubic functions have the form f(x) equals ax cubed plus bx squared plus cx plus d. Exponential functions like f(x) equals a to the power of x grow or decay rapidly. Logarithmic functions are the inverse of exponential functions. And trigonometric functions like sine and cosine are periodic. Functions can have special properties: a one-to-one function means each output has exactly one input, allowing for an inverse function. An onto function means every element in the codomain is used as an output. These different types of functions have various applications in mathematics, science, engineering, and economics.
Functions have numerous real-world applications. In physics, functions model position, velocity, and acceleration. In economics, they represent cost, revenue, and profit relationships. In biology, functions model population growth, as shown in our graph with linear, exponential, and logistic growth models. In computer science, functions are fundamental to algorithms and programming. To summarize what we've learned: a function is a relation that maps each input to exactly one output. The domain is the set of all possible inputs, while the range is the set of all possible outputs. We can use the vertical line test to identify if a graph represents a function. Functions are powerful mathematical tools that help us model and understand relationships in virtually every field of study. They allow us to make predictions, analyze patterns, and solve complex problems across disciplines.