Function Value Transformation

The problem asks us to determine the new value of a given function \(f(x, y)\) when its input variables \(x\) and \(y\) undergo specific changes. We are initially provided with the function and its current value:

• The original function is defined as: \(f(x, y) = x^n y^m\).
• The initial value of this function is given as \(P\).

From this information, we can establish the fundamental relationship: \(P = x^n y^m\).

Variable Changes Analysis

The problem specifies how the independent variables \(x\) and \(y\) are modified from their original states:

• The variable \(x\) is explicitly stated to be doubled. This means its new value will be \(2x\).
• The variable \(y\) is stated to be halved. This implies its new value will be \(\frac{y}{2}\).

Calculating New Function Value Step-by-Step

To find the new value of the function, let's substitute these modified values of \(x\) and \(y\) back into the original function definition. Let \(f_{new}\) represent this new function value.

1. Substitute the modified variables into the function: The original function is \(f(x, y) = x^n y^m\). By replacing \(x\) with \((2x)\) and \(y\) with \(\left(\frac{y}{2}\right)\), we get: \[f_{new} = (2x)^n \left(\frac{y}{2}\right)^m\]
2. Apply the exponent rules: Recall the exponent rules: \((ab)^p = a^p b^p\) and \(\left(\frac{a}{b}\right)^p = \frac{a^p}{b^p}\). Applying these rules to our expression: \[f_{new} = (2^n x^n) \left(\frac{y^m}{2^m}\right)\] This simplifies to: \[f_{new} = 2^n \cdot x^n \cdot \frac{y^m}{2^m}\]
3. Rearrange the terms: Group the constant terms (powers of 2) together and the variable terms (\(x^n y^m\)) together: \[f_{new} = \left(\frac{2^n}{2^m}\right) \cdot (x^n y^m)\]
4. Simplify the powers of 2: Using another exponent rule: \(\frac{a^p}{a^q} = a^{p-q}\). Apply this to \(\left(\frac{2^n}{2^m}\right)\): \[f_{new} = 2^{n-m} \cdot (x^n y^m)\]
5. Substitute back the initial value \(P\): We initially established that \(P = x^n y^m\). Substitute \(P\) back into our simplified expression for \(f_{new}\): \[f_{new} = 2^{n-m} P\]

Conclusion on New Function Value

Therefore, when the variable \(x\) is doubled and the variable \(y\) is halved, the new value of the function \(f(x, y)\) is found to be \(2^{n-m}P\).