A function of several variables assigns one output to each allowed combination of two or more inputs. In calculus, this lets us model quantities that depend on position, time, temperature, pressure, or other changing factors. A function like z = f(x, y) can describe a height above a map, the temperature on a metal plate, or the cost of producing two products.
These functions matter because real systems usually depend on more than one input at a time.
For a function of two variables, the graph is often a surface in three-dimensional space, where each point (x, y) in the domain gives a height z. For a function of three variables, such as w = f(x, y, z), the graph would need four dimensions, so we often study level surfaces or slices instead. The domain tells which input combinations are allowed, and restrictions often come from square roots, denominators, logarithms, or real-world limits.
Evaluating a multivariable function means substituting all input values carefully and simplifying the resulting expression.
Key Facts
- A function of two variables has the form z = f(x, y).
- A function of three variables has the form w = f(x, y, z).
- The domain is the set of all input pairs or triples for which the function is defined.
- The graph of z = f(x, y) is a surface in 3D space.
- To evaluate f(a, b), substitute x = a and y = b into the formula.
- Level curves satisfy f(x, y) = c and show where the surface has constant height c.
Vocabulary
- Function of several variables
- A rule that assigns one output value to each allowed ordered pair, triple, or larger set of inputs.
- Domain
- The complete set of input values for which a function is defined.
- Surface
- The three-dimensional graph formed by plotting points (x, y, z) where z = f(x, y).
- Level curve
- A curve in the input plane where a function of two variables has one constant output value.
- Evaluation
- The process of finding a function output by substituting specific input values into the formula.
Common Mistakes to Avoid
- Treating z = f(x, y) like a single-variable function is wrong because the output depends on both x and y, not just one input.
- Ignoring domain restrictions is wrong because expressions such as square roots, denominators, and logarithms may make some input pairs invalid.
- Confusing the domain with the graph is wrong because the domain lies in the input space, while the graph of z = f(x, y) lies in three-dimensional space.
- Substituting only one variable during evaluation is wrong because every input variable must be replaced with its given value before simplifying.
Practice Questions
- 1 Let f(x, y) = x^2 + 3y. Find f(2, -1).
- 2 Find the domain of f(x, y) = sqrt(9 - x^2 - y^2). Describe it in words and with an inequality.
- 3 A surface z = f(x, y) represents temperature on a flat metal plate. Explain what a level curve f(x, y) = 50 means in this context.