What is a function? A function is a relation between a set of inputs and a set of possible outputs where each input is related to exactly one output. The key rule is that for every input value, there is exactly one corresponding output value. In this diagram, we can see inputs on the left and outputs on the right. Notice how different inputs like 2 and 3 can map to the same output, which is 5. However, each input has exactly one arrow coming out of it, meaning each input maps to exactly one output. This is what makes it a function.
Let's look at function notation and examples. We typically write a function as f of x equals y, where f is the name of the function, x is the input or independent variable, and y is the output or dependent variable. The expression f of x means 'the value of function f at input x'. Here's a simple example: the function f of x equals x squared. When we input a value like x equals 2, the function gives us the output f of 2 equals 4. As we change the input value, the output changes according to the function rule. Notice how each input x corresponds to exactly one output on the curve, which is the defining property of a function.
Let's explore how to determine if a relation is a function or not using the vertical line test. If any vertical line intersects a graph at more than one point, 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. This confirms it's a function - each input has exactly one output. 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 for most x-values. This means some inputs have two possible outputs, violating the definition of a function. The vertical line test is a simple visual way to check if a graph represents a function.
Let's explore some common types of functions. 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 appear as straight lines on a graph. Quadratic functions follow the form f of x equals a x squared plus b x plus c, creating parabolas. Exponential functions like f of x equals a to the power of x grow or decay at rates proportional to their current value. Logarithmic functions are the inverse of exponential functions. And trigonometric functions like sine and cosine model periodic phenomena. Each type of function has distinct characteristics and applications in various fields. For example, linear functions model constant rates of change, while exponential functions model population growth or compound interest.
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 use the notation f of x equals y, where f is the function name, x is the input, and y is the output. The vertical line test helps us determine if a graph represents a function - if any vertical line intersects the graph more than once, it's not a function. We've explored various types of functions including linear, quadratic, exponential, logarithmic, and trigonometric functions, each with unique properties and applications. Functions are fundamental mathematical tools that help us model relationships in science, engineering, economics, and countless aspects of daily life. They allow us to describe how one quantity depends on another, making them essential for understanding and predicting change.