Problem 12 - Calculus and its applications

Question

Determine the domain of each function of two variables. $$ h(x, y)=x e^{\sqrt{y}} $$

Step-by-Step Solution

Answer

The domain is all \( (x, y) \) such that \( y \geq 0 \).

Step 1: Analyze the Function's Components

To determine the domain of the function \( h(x, y) = x e^{\sqrt{y}} \), we need to identify any restrictions on the variables \( x \) and \( y \). This function involves two primary components: \( x \) which is unrestricted, and \( e^{\sqrt{y}} \), which is dependent on \( y \).

Step 2: Consider Restrictions from the Exponential Component

The expression \( e^{\sqrt{y}} \) involves the square root of \( y \). The square root function, \( \sqrt{y} \), is real and defined only for \( y \geq 0 \). Hence, \( y \) must be equal to or greater than zero to ensure that \( \sqrt{y} \) is real.

Step 3: Determine the Domain from Combined Conditions

Since the exponential function \( e^{\sqrt{y}} \) requires \( y \geq 0 \) and \( x \) is unrestricted, the domain of the function \( h(x, y) \) is constrained by \( y \geq 0 \). Hence, the domain encompasses all real \( x \) and \( y \) values that satisfy \( y \geq 0 \).

Key Concepts

Functions of Two Variables

Functions of two variables, such as the function \( h(x, y) = x e^{\sqrt{y}} \), are mathematical expressions that depend on two different input values. In this case, \( x \) and \( y \) serve as the variables, and the function maps pairs of these variables to a single output value.

• Each variable in a function of two variables can affect the output in distinct ways.
• The role each variable plays will depend on how it is integrated into the function.
• In \( h(x, y) \), one part of the function, \( x \), is linear and unrestricted, impacting the function output linearly.
• Meanwhile, \( e^{\sqrt{y}} \) is exponential in nature, offering unique characteristics that depend on the value of \( y \).

Understanding these nuances helps in deciphering how changes in \( x \) and \( y \) result in changes to \( h(x, y) \). Recognizing any variable restrictions is also critical in determining the function's domain, that is, the set of permissible \( x \) and \( y \) values.

Domain of a Function

The domain of a function is essentially the set of all possible input values that allow the function to produce a real output. For a function of two variables, it is crucial to consider any mathematical operations that can affect the permissible inputs.
When evaluating \( h(x, y) = x e^{\sqrt{y}} \), we focus on the expression \( e^{\sqrt{y}} \).

• Since the exponential component involves a square root, we must ensure \( \sqrt{y} \) is defined.
• Mathematically, the square root function is only real when \( y \geq 0 \); thus, this places a condition on our domain related to \( y \).
• On the other hand, \( x \) remains unrestricted as there are no operations involving \( x \) that limit its value.

Therefore, the domain of \( h(x, y) \) is described by all real numbers \( x \), with \( y \) being at least zero (\( y \geq 0 \)). This means any pair \((x, y)\) that satisfies \( y \geq 0 \) can be used within the function.

Exponential Functions

Exponential functions are fascinating due to their rapid rate of growth and distinct properties. They are commonly expressed in the form \( e^x \), where \( e \) is the mathematical constant approximately equal to 2.718.
In the function \( h(x, y) = x e^{\sqrt{y}} \), the exponential component \( e^{\sqrt{y}} \) connects exponential growth with the variable \( y \) through a square root.

• Exponential functions amplify changes in their exponent, leading to faster increases or decreases in their output compared to linear functions.
• In this function, as \( y \) increases, \( \sqrt{y} \) increases, leading to larger values of \( e^{\sqrt{y}} \).
• This component is always positive as exponential functions never cross the horizontal axis, further influencing the behavior of the function \( h(x, y) \).
• Understanding this effect can provide insights into how \( h(x, y) \) behaves as \( y \) changes, especially since \( x \) affects the function linearly unlike \( e^{\sqrt{y}} \).

Thus, exponential functions' unique characteristics significantly influence functions like \( h(x, y) \), dictating how different variables can impact the result and forming an important part of understanding the function's complete behavior.