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 characteristic of a function is that each input value maps to exactly one output value. For example, in the function f of x equals zero point five x squared, when we input the value 2, we get exactly one output, which is 2.
Function notation is typically written as f of x equals y, where f is the name of the function, x is the input value from the domain, and y is the output value in the range. Common examples of functions include f of x equals x squared, which is a parabola, g of x equals two x plus three, which is a linear function, and h of x equals sine of x, which is a trigonometric function. Each of these functions takes an input value and produces exactly one output value according to its rule.
The domain of a function 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 applying the function. Let's look at the square root function as an example. For f of x equals the square root of x, the domain is restricted to non-negative numbers, since we cannot take the square root of a negative number in the real number system. So the domain is x is greater than or equal to zero. The range is also restricted to non-negative numbers, as the square root of a non-negative number is always non-negative. Therefore, the range is y is greater than or equal to zero.
Function composition is the process of applying one function to the result of another. We write f composed with g of x equals f of g of x, which means we first apply function g to x, and then apply function f to that result. Functions can also be transformed in various ways. Common transformations include vertical shifts, where we add a constant to the function, resulting in the graph moving up or down. Horizontal shifts occur when we replace x with x minus c, moving the graph left or right. Vertical stretches or compressions happen when we multiply the function by a constant. And reflection across the x-axis occurs when we negate the function. These transformations allow us to create new functions from existing ones.
To summarize what we've learned about functions: A function is a special type of relation where each input maps to exactly one output. Functions are typically written using the notation f of x equals y, where x is the input from the domain and y is the output in the range. The domain represents all valid input values, while the range consists of all possible output values. Functions can be transformed through shifts, stretches, reflections, and can be composed with other functions to create new functions. Functions are fundamental mathematical tools used extensively in science, engineering, economics, and data analysis to model relationships between variables.