Multivariable Functions

Sometimes we think of functions as little machines that turn an input number into an output number.

The function

f(x) = 2x

is a doubling factory.

But to a mathematician, a function doesn't need to take one input and give one output. Some functions take two numbers as input and give one number as output.

The function

f(x, y) = x + 2y

combines two inputs into one output.

Multivariable Equations

Evaluating a function like $f(x, y) = x^2 + y^2$ works the same way as evaluating a familiar 2D function. To find $f(3, 4)$ , you replace each x with 3 and each y with 4.

f(, ) = ² + ² = 25

Try changing the inputs!

Watch out! Unlike what we're used to, y is an input to this function, not the output.

Let's practice!

If $f(x, y) = 5x - y^2$ then $f(3, 2) =$ ?

Multivariable Tables

The table for a multivariable function has two (or more) inputs and one output. For example, consider this function that calculates the cost of a rental car at $25/day + $0.14/mile.

Input #1 →

Input #2 →

Output →

x Days	1	2	1	3	4
y Miles	30	65	24	92	105
$f(x, y)$ Price ($)	29.2	59.1	28.36	87.88	114.7

Let's practice!

Complete the following table for the function $f(a, b) = 2ab$

a	b	$f(a, b)$
0	3
1	4
2	5

a	b	$f(a, b)$
0	3
1	4
2	5

a	b	$f(a, b)$
0	3
1	4
2	5

Multivariable Graphs

A normal function with one input and one output is graphed on two axes, usually x and y.

y = f(x)

With a multivariable function, we have more than one input, so we need to add a third axis. This makes a 3D graph.

z = f(x, y)