When you start learning about functions, the left hand side is usually y, and then on the right hand side we build up more complicated expressions from linear, to quadratic and cubic expressions, etc. using a variable, x.

One day, the left hand side changes from y to something like f(x), or g(x)...what happened?

f(x) notation introduces the idea that functions have inputs, use the inputs in a calculation, and produce an output. In this case f only has one input. The inputs are the values in the parentheses. When the input is "x" the output of the function is an expression in x. When the input is anything other than "x" you replace "x" by the input value and calculate the result.

For example, f(x) = 5(x^2) + (3x) -7 shows the output of a function f with one input when it is given the input x i.e. f(x). If it is given the input 8 it is written as f(8) and f(8) = 5(8^2) +(3*8) -7 = 337

It's just a new notation, but it is powerful! The notation allows us to describe functions that depend on multiple variables. For example f(x, y) describes a function that depends on two inputs; you need to know both of their values AND what it does with them in order to calculate f's output.

Right hand side variables like x, y, z, t are called independent variables, because they are free to take any values in their domain. We often use x,y to define the position of an object in 2D space, x,y,z for an object in 3D space, and "t" is usually time. P(x,y,z,t) can define the position of an object in space at any moment in time. It is powerful!

The notation also allows us to create a function of a function. For example f(g(6)) means "calculate the output of g when its input is 6, and then use that output as the input into f. If f(x) = 2x + 3 and g(x) = x^2 then g(3) = 9 and f(g(3)) = f(9) = 21

Example. The area(A) of a carpet depends on its width(W) and length(L), so we can write A(W, L) = WL as an expression with two inputs W and L, which outputs the areal Then A(9,12) outputs the area of a carpet 9 feet wide and 12 feet long, i.e. 108 square feet.

Similarly, the volume (V) of a rectangular box with height (H) can be written as V(W, L, H) = WLH. So V(30, 6, 8) can describe the volume of a container truck that is 30'**6'*8'. *The output will be 1,440 cubic feet.* *

When f, g, or h only depend on one variable, like x, it is sometimes convenient in 2D graphing to call it just __y__, but you can see how the new notation enables us to build much more complicated functions.

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