Question

Determining if a Function Is Homogeneous In Exercises \(67-74,\) determine whether the function is homogeneous, and if it is, determine its degree. A function \(f(x, y)\) is homogeneous of degree \(n\) if \(f(t x, t y)=t^{n} f(x, y)\) $$ f(x, y)=2 \ln \frac{x}{y} $$

Short answer

The function is not homogeneous.

Step-by-step solution

Plug in \(t \cdot x\) and \(t \cdot y\) into the equation

We replace \(x\) with \(t \cdot x\) and \(y\) with \(t \cdot y\) in the original function:\[f(t \cdot x, t \cdot y) = 2 \ln \frac{t \cdot x}{ t \cdot y}\]

Simplify the new function

We then simplify this function:\[f(t \cdot x, t \cdot y) = 2 \ln \left(t \cdot \frac{x}{y}\right)\],which further becomes,\[f(t \cdot x, t \cdot y) = \ln \left(t^2 \cdot \frac{x}{y}\right)\]

Compare the new function with \(t^n \cdot f(x, y)\)

The simplified function does not take the form \(t^n \cdot f(x,y)\), instead of \(t\) raised to some power multiplied by the original function, we get a function that involves the square of \(t\) inside the logarithm. This shows that the function is not homogeneous.

Key concepts

Degree of Homogeneity

In the realm of mathematics, particularly in multivariable calculus, a function's degree of homogeneity is a pivotal concept. To understand this, consider a function, say \(f(x, y)\), which is said to be homogeneous of degree \(n\) if the scaling of variables \(t \, x\) and \(t \, y\) results in \(t^n\) times the original function. This can be elegantly represented as \(f(t \, x, t \, y) = t^n \, f(x, y)\).

When applied, it means that multiplying each argument of the function by a constant \(t\), scales the function by \(t^n\). Think of it as a way to measure how the function "behaves" under scaling.

In practical terms, determining the degree of homogeneity helps us understand the characteristics and scaling properties of functions, especially when they represent real-world phenomena or mathematical models.

Logarithmic Functions

Logarithmic functions are unique functions based on exponential inverses. In simpler terms, if an exponential function like \(b^y = x\) is given, the logarithmic function finds the exponent \(y\).

For example, \(\ln(x)\) equals the power to which the base \(e\) must be raised to produce \(x\). These functions are pivotal because they help manage scale. They transform multiplication into addition, growing the calculation scope without heavy computation.

Given the exercise function \(f(x, y) = 2 \ln \frac{x}{y}\), logarithmic properties allow simplification and scaling checks, like between constants and variables. The behavior under transformation, such as those in homogeneity tests, highlights the special considerations when \(\ln\) structures are involved.

Multivariable Calculus

Multivariable calculus is a natural expansion of traditional calculus into functions with more than one variable. It explores how functions like \(f(x, y)\) change across multiple dimensions and is crucial for applications in fields like physics and economics.

With functions like \(f(x, y) = 2 \ln \frac{x}{y}\), multivariable calculus investigates changes not just along one dimension but considers influences from both \(x\) and \(y\). These changes are represented by gradients, derivatives, and other tools unique to multivariable scenarios.

The main goal in this discipline is to achieve a comprehensive realization of how variable changes influence the entire system, offering deeper insights into optimization, integration, and dynamic behavior.

Function Transformation

Transformation in functions involves altering their form through translations, scaling, rotations, or reflections. For the function \(f(x, y) = 2 \ln \frac{x}{y}\), transformation means examining how the form changes when inputs \(x\) and \(y\) are manipulated.

• Scaling: When inputs \(x\) and \(y\) are changed by a factor \(t\)
• Reflection: Inverting variables \(x\) or \(y\)

By applying transformations, one can explore how the output changes, which is central to understanding and predicting behavior.

These transformations are crucial for evaluating homogeneity. The exercise does just that, determining whether scaled inputs align with a uniform transformation expression \(t^n \cdot f(x, y)\). In this function, transformations expose its resistance to standard homogeneity rules, leading to a deeper grasp of composition and functional behavior.