You can graph circles, ellipses, lines and parabolas and represent all these by equations in math. However, not all these equations are functions. In math, a function is an equation with only one output for each input. In the case of a circle, one input can give you two outputs - one on each side of the circle. Thus, the equation for a circle is not a function and you cannot write it in function form.
Apply the vertical line test to determine if your equation is a function. If you can move a vertical line along the x-axis and only intersect one y at a time, your equation is a function as it follows the only one output for each input rule.
Solve your equation for y. For instance, if your equation is y -6 = 2x, add 6 to both sides to get y = 2x + 6.
Decide on a name for your function. Most functions use a one-letter name such as f, g or h. Determine what variable your function depends upon. In the example of y = 2x + 6, the function changes as the value of x changes, so the function is dependent upon x. The left side of your function is the name of your function followed by the dependent variable in parenthesis, f(x) for the example.
Write your function. The example becomes f(x) = 2x + 6.
You write functions with the function name followed by the dependent variable, such as f(x), g(x) or even h(t) if the function is dependent upon time. You read the function f(x) as "f of x" and h(t) as "h of t". Functions do not have to be linear. The function g(x) = -x^2 -3x + 5 is a nonlinear function. The equation is nonlinear because of the square of x, but it is still a function because there is only one answer for every x. When evaluating a function for a specific value, you place the value in the parenthesis rather than the variable. For the example of f(x) = 2x + 6, if you wish to find the value when x is 3, you write f(3) = 12 since 2 times 3 plus 6 is 12. Similarly, f(0) = 6 and f(-1) = 4.
Do not confuse function names with multiplication. Function f(x) is not variable f times variable x. Function f(x) is a function named f that depends upon x.