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Multivariable Functions

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

5
The function f(x)=2xf(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+2yf(x, y) = x + 2y combines two inputs into one output.

Multivariable Equations

Evaluating a function like f(x,y)=x2+y2f(x, y) = x^2 + y^2 works the same way as evaluating a familiar 2D function. To find f(3,4)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)=5xy2f(x, y) = 5x - y^2 then f(3,2)=f(3, 2) = ?

Answer:

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
12134
y
Miles
30652492105
f(x,y)f(x, y)
Price ($)
29.259.128.3687.88114.7

Let's practice!

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

Answer:
a
b
f(a,b)f(a, b)
03
14
25

Multivariable Graphs

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

y=f(x)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)z = f(x, y)