The Main Idea 

A function of two variables takes an ordered pair [latex](x, y)[/latex] as input and produces a single output [latex]z[/latex]. It’s written as [latex]z = f(x, y)[/latex].

Instead of mapping one number to another like [latex]y = f(x)[/latex], you map pairs of numbers to a single output. Each point [latex](x, y)[/latex] in the domain produces exactly one value [latex]z[/latex].

Domain and Range

• Domain: The set of all ordered pairs [latex](x, y)[/latex] for which the function is defined
• Range: The set of all possible output values [latex]z[/latex]

Finding the Domain

Look for the same restrictions as single-variable functions:

• Square roots need non-negative expressions: [latex]\sqrt{9 - x^2 - y^2}[/latex] requires [latex]9 - x^2 - y^2 \geq 0[/latex]
• Denominators can’t be zero
• Logarithms need positive arguments

The difference? These restrictions now define regions in the [latex]xy[/latex]-plane, not intervals on a line.

Finding the Range

Pick a value [latex]z[/latex] and ask: can I find [latex](x, y)[/latex] that makes [latex]f(x, y) = z[/latex]? The range includes all [latex]z[/latex] values where the answer is yes.

Problem-Solving Strategy

Identify restrictions on [latex]x[/latex] and [latex]y[/latex] to find the domain. Test boundary and interior points to determine the range. Sketch the domain region in the [latex]xy[/latex]-plane when helpful.

The Main Idea 

The graph of [latex]z = f(x, y)[/latex] is a surface in three-dimensional space. Each point [latex](x, y)[/latex] in the domain gets a height [latex]z[/latex], creating ordered triples [latex](x, y, z)[/latex].

Picture the [latex]xy[/latex]-plane lying flat. For each point [latex](x, y)[/latex] in the domain, the function assigns a [latex]z[/latex]-value. Positive [latex]z[/latex] means the point sits above the plane. Negative [latex]z[/latex] means it sits below. All these points together form a surface.

Common Surfaces

Hemisphere: [latex]g(x, y) = \sqrt{9 - x^2 - y^2}[/latex]

• Domain: [latex]x^2 + y^2 \leq 9[/latex] (disk of radius 3)
• Range: [latex]0 \leq z \leq 3[/latex]
• Shape: Upper half of a sphere

Paraboloid: [latex]f(x, y) = x^2 + y^2[/latex]

• Domain: All of [latex]\mathbb{R}^2[/latex]
• Range: [latex]z \geq 0[/latex]
• Shape: Bowl opening upward

Understanding the Shape

Look at cross-sections by holding one variable constant:

• Set [latex]x = 0[/latex]: You get [latex]z = f(0, y)[/latex], often a familiar curve
• Set [latex]y = 0[/latex]: You get [latex]z = f(x, 0)[/latex], another familiar curve
• Set [latex]z = c[/latex]: You get level curves showing where the surface has height [latex]c[/latex]

Problem-Solving Strategy

Identify the domain. Find key cross-sections. Determine what happens at boundaries. Recognize familiar surface types like spheres, paraboloids, or planes.

The Main Idea 

A level curve shows all points where a function [latex]f(x, y)[/latex] equals a constant value [latex]c[/latex]. It’s like a contour line on a topographical map—points on the same curve have the same “height.”

For a function [latex]z = f(x, y)[/latex] and a value [latex]c[/latex], the level curve is the set of points satisfying [latex]f(x, y) = c[/latex]. Each different value of [latex]c[/latex] gives a different curve in the [latex]xy[/latex]-plane.

Example: Hemisphere

For [latex]g(x, y) = \sqrt{9 - x^2 - y^2}[/latex], set [latex]g(x, y) = 2[/latex]:

• Solve: [latex]\sqrt{9 - x^2 - y^2} = 2[/latex]
• Square: [latex]9 - x^2 - y^2 = 4[/latex]
• Simplify: [latex]x^2 + y^2 = 5[/latex]

This is a circle of radius [latex]\sqrt{5}[/latex]. Different [latex]c[/latex] values give concentric circles.

Reading Level Curves

• Curves close together: Function changes rapidly (steep terrain)
• Curves far apart: Function changes slowly (gentle slope)

Contour Maps

A contour map shows multiple level curves together. It gives a two-dimensional view of a three-dimensional surface.

Vertical Traces

While level curves live in the [latex]xy[/latex]-plane, vertical traces show slices through the surface:

• Set [latex]x = a[/latex]: Get a curve in the [latex]yz[/latex]-plane
• Set [latex]y = b[/latex]: Get a curve in the [latex]xz[/latex]-plane

Vertical traces help visualize the surface’s shape from different angles.

Problem-Solving Strategy

Choose a value [latex]c[/latex]. Set [latex]f(x, y) = c[/latex]. Solve for the relationship between [latex]x[/latex] and [latex]y[/latex]. Repeat for several [latex]c[/latex] values to build a contour map.

The Main Idea 

Functions can have three or more independent variables. A function [latex]f(x, y, z)[/latex] takes a point in three-dimensional space and maps it to a single output value.

Finding the Domain

Use the same restrictions as before:

• Square roots need nonnegative radicands
• Denominators can’t be zero
• Logarithms need positive arguments

Example: [latex]f(x, y, z) = \frac{3x - 4y + 2z}{\sqrt{9 - x^2 - y^2 - z^2}}[/latex]

The denominator requires [latex]9 - x^2 - y^2 - z^2 > 0[/latex], or [latex]x^2 + y^2 + z^2 < 9[/latex]. This is a ball of radius 3 (without the surface).

Level Surfaces

For functions of three variables, level curves become level surfaces. A level surface is the set of all points where [latex]f(x, y, z) = c[/latex] for some constant [latex]c[/latex].

The Pattern

As you add variables, level sets increase in dimension:

• Two variables → level curves (1D)
• Three variables → level surfaces (2D)
• Four variables → level hypersurfaces (3D)