Think of a function as a simple machine. You put something in (an input), and it gives you something back (an output). The rule is that for every one input, you get exactly one output. It's a consistent, predictable relationship.

We usually write functions using a special notation. If we have a function named $f$, we write $f(x)$ to represent the output when the input is $x$. You can read $f(x)$ as "f of x."

For example, a function that doubles the input can be written as:

Every function has two important sets of values associated with it: its domain and its range.

The domain is the complete set of all possible inputs a function can accept. The range is the complete set of all possible outputs it can produce.

Consider our doubling function, $f(x) = 2x$. If we can plug in any real number, its domain is all real numbers. Since doubling any real number results in another real number, its range is also all real numbers.

But sometimes domains are restricted. For the function $g(x) = \sqrt{x}$, we can't take the square root of a negative number (and get a real result). So, its domain is all non-negative numbers, $x \ge 0$. The outputs are also never negative, so its range is also all non-negative numbers, $g(x) \ge 0$.

Different types of functions have distinct formulas and create unique shapes when graphed. Let's look at some of the most common ones.

Quadratic Functions These have the form $f(x) = ax^2 + bx + c$. Their graphs are U-shaped curves called parabolas. If $a$ is positive, the parabola opens upwards. If $a$ is negative, it opens downwards.

Rational Functions A rational function is a fraction of two polynomials, like $f(x) = \frac{P(x)}{Q(x)}$. Their graphs can have breaks called asymptotes, which are lines the graph approaches but never touches. These occur where the denominator is zero.

Exponential and Logarithmic Functions Exponential functions, like $f(x) = 2^x$, model rapid growth or decay. Logarithmic functions, like $g(x) = \log_2(x)$, are their inverses. They grow very slowly.

Once you know the basic shape of a function's graph, you can move it around without replotting every point. These movements are called transformations.

For example, if we know the graph of $f(x) = x^2$ is a parabola centered at the origin $(0,0)$, then the graph of $g(x) = (x-3)^2 + 5$ is the exact same parabola, just shifted 3 units to the right and 5 units up.