91Ó°ÊÓ

Two-variable functions are mathematical expressions that depend on two independent variables. In the case of the function given, \( g(x, y) = e^{xy} \), both \( x \) and \( y \) are the independent variables that influence the function's output. This means that the value of the function is determined by the product \( xy \).

Such functions are represented on a three-dimensional graph, where usually the two variables (\( x \) and \( y \)) define the horizontal plane, and the function value, \( g(x, y) \), projects onto the vertical axis.

• Familiarize with how changing one or both variables affects the function value.
• Understand how contour or level curves offer a two-dimensional perspective of the function's behavior.

These level curves occur where the function equals a constant value, \( c \), and help visualize how \( x \) and \( y \) interact to reach that value.

Exponential functions are characterized by having a variable as an exponent. In the two-variable function \( g(x, y) = e^{xy} \), the expression involves the base \( e \) raised to the power of \( xy \).

One key property of exponential functions is their consistent growth rate. They increase rapidly because each increment in the exponent results in a proportional increase in the function's value.

• Understand that exponential functions can model real-life growth scenarios such as population or cooling processes.
• Recognize the logarithm as the inverse operation to exponentiation, which is useful in solving equations involving exponentials.

By setting the exponential function equal to a constant (like \( c = \frac{1}{2} \) or \( c = 3 \)), one can use logarithms to solve for the variable terms, further defining the level curves that are the focus of this problem.

Hyperbolas are a type of conic section, which appear as open curves that approach two intersecting asymptotes. When level curves of the exponential function \( e^{xy} \) are explored, certain values of \( c \) lead to equations of the form \( xy = \text{constant} \), which represent hyperbolas.

These hyperbolas occur because the equations take the form of \( xy = c \), describing curves symmetrical about both the \( x \)- and \( y \)-axes. They demonstrate how the values of \( x \) and \( y \) correlate inversely to maintain the equation.

• The shape of a hyperbola is defined by the equation's constant; a negative constant results in hyperbolas positioned differently than a positive one.
• The curves never intersect their asymptotes, elucidating how hyperbolas spread over the plane.

This concept explains why, in this exercise, the level curves for \( c = \frac{1}{2} \) and \( c = 3 \) transform into hyperbolas representing different levels of function values.