Algebra Study Guide

In mathematics, a function is a special type of relationship between two sets of numbers, often called the input (or domain) and the output (or range). For every single input value, there is exactly one output value. Think of it like a machine: you put something in, and the machine processes it to give you a unique result. If you put the same thing in again, you'll get the exact same result out. Functions are often represented using 'f(x)' notation, where 'f' is the name of the function and 'x' is the input variable. For example, if we have a function f(x) = 2x + 1, it means that for any input 'x', the function takes 'x', multiplies it by 2, and then adds 1 to get the output. To find the output when the input is, say, 3, we substitute 3 for 'x': f(3) = 2(3) + 1 = 6 + 1 = 7. So, the input 3 gives an output of 7. Functions can also be represented by graphs, where the x-axis represents the input and the y-axis represents the output. A key test for a function from its graph is the 'vertical line test': if any vertical line intersects the graph at more than one point, it is not a function, because a single x-value would have multiple y-values. Understanding functions is fundamental in algebra and many applications of mathematics.

A function is a rule that assigns each input value (from its domain) to exactly one output value (in its range). We often write functions using notation like f(x) = 2x + 1, where 'f' is the name of the function and 'x' is the input variable. To find the output for a specific input, say x=3, we substitute 3 into the function: f(3) = 2(3) + 1 = 7.

Composite functions involve applying one function after another. If we have functions f(x) and g(x), then fg(x) means we first apply function g to x, and then apply function f to the result. So, fg(x) = f(g(x)). Conversely, gf(x) = g(f(x)). The order matters significantly in composite functions.

An inverse function, denoted by f⁻¹(x), 'undoes' the action of the original function f(x). If f(a) = b, then f⁻¹(b) = a. Graphically, the inverse function is a reflection of the original function across the line y = x. To find the inverse function, you typically follow these steps: 1. Replace f(x) with y. 2. Swap x and y. 3. Rearrange the equation to make y the subject again. 4. Replace y with f⁻¹(x). It's important to remember that not all functions have an inverse that is also a function. For an inverse function to exist over its entire domain, the original function must be one-to-one (meaning each output corresponds to exactly one input). If it's not one-to-one, we may need to restrict the domain of the original function for its inverse to be a valid function.