Multivariable Functions: Understanding Domains, Range, and Level Curves, Study notes of Mathematics

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Math 114-004, Fall 2009

Tong Zhu

Department of Mathematics University of Pennsylvania

October 16, 2009

Multivariable Functions

Multivariable Functions

What is a multivariable function?

f : (x 1 , x 2 , ..., xn) → y , where x 1 , x 2 , ..., xn, y are all real numbers

Multivariable Functions

What is a multivariable function?

f : (x 1 , x 2 , ..., xn) → y , where x 1 , x 2 , ..., xn, y are all real numbers

For example, a two-variable function f maps an ordered pair of real numbers (x, y ) to a unique real number, say z.

Multivariable Functions

What is a multivariable function?

f : (x 1 , x 2 , ..., xn) → y , where x 1 , x 2 , ..., xn, y are all real numbers

For example, a two-variable function f maps an ordered pair of real numbers (x, y ) to a unique real number, say z.

Example 1 f (x, y ) =

√x−y x+y. What are the domain and range of this two variable function?

Multivariable Functions

What is a multivariable function?

f : (x 1 , x 2 , ..., xn) → y , where x 1 , x 2 , ..., xn, y are all real numbers

For example, a two-variable function f maps an ordered pair of real numbers (x, y ) to a unique real number, say z.

Example 1 f (x, y ) =

√x−y x+y. What are the domain and range of this two variable function?

I (^) Domain:

Multivariable Functions

What is a multivariable function?

f : (x 1 , x 2 , ..., xn) → y , where x 1 , x 2 , ..., xn, y are all real numbers

For example, a two-variable function f maps an ordered pair of real numbers (x, y ) to a unique real number, say z.

Example 1 f (x, y ) =

√x−y x+y. What are the domain and range of this two variable function?

I (^) Domain:x − y ≥ 0 and x + y 6 = 0 written as D = {(x, y )|x ≥ y and x + y 6 = 0, x, y ∈ R}

Multivariable Functions

What is a multivariable function?

f : (x 1 , x 2 , ..., xn) → y , where x 1 , x 2 , ..., xn, y are all real numbers

For example, a two-variable function f maps an ordered pair of real numbers (x, y ) to a unique real number, say z.

Example 1 f (x, y ) =

√x−y x+y. What are the domain and range of this two variable function?

I (^) Domain:x − y ≥ 0 and x + y 6 = 0 written as D = {(x, y )|x ≥ y and x + y 6 = 0, x, y ∈ R} Can you sketch the region of this domain?

Multivariable Functions

What is a multivariable function?

f : (x 1 , x 2 , ..., xn) → y , where x 1 , x 2 , ..., xn, y are all real numbers

For example, a two-variable function f maps an ordered pair of real numbers (x, y ) to a unique real number, say z.

Example 1 f (x, y ) =

√x−y x+y. What are the domain and range of this two variable function?

I (^) Domain:x − y ≥ 0 and x + y 6 = 0 written as D = {(x, y )|x ≥ y and x + y 6 = 0, x, y ∈ R} Can you sketch the region of this domain? I (^) Range: R = R

Example 2 Find the domain and range of f (x, y , z) = e

z−x^2 −y (^2).

Example 2 Find the domain and range of f (x, y , z) = e

z−x^2 −y (^2).

Solution:

I (^) Domain: z ≥ x^2 + y 2

Example 2 Find the domain and range of f (x, y , z) = e

z−x^2 −y (^2).

Solution:

I (^) Domain: z ≥ x^2 + y 2 ⇒ D = {(x, y , z) ∈ R^3 |x^2 + y 2 ≤ z}

Example 2 Find the domain and range of f (x, y , z) = e

z−x^2 −y (^2).

Solution:

I (^) Domain: z ≥ x^2 + y 2 ⇒ D = {(x, y , z) ∈ R^3 |x^2 + y 2 ≤ z} Where is this domain? I (^) Range:

Example 2 Find the domain and range of f (x, y , z) = e

z−x^2 −y (^2).

Solution:

I (^) Domain: z ≥ x^2 + y 2 ⇒ D = {(x, y , z) ∈ R^3 |x^2 + y 2 ≤ z} Where is this domain? I (^) Range: Let w = f (x, y , z). R = {w ∈ R|w ≥ 1 }