Problem 6 - Contemporary Precalculus

Question

Find \((f g)(x),(f / g)(x),\) and \((g / f)(x)\) $$f(x)=4 x^{2}+x^{4}, \quad g(x)=\sqrt{x^{2}+4}$$

Step-by-Step Solution

Answer

Question: Calculate the product, quotient, and reverse quotient of the functions f(x) = 4x^2 + x^4 and g(x) = √(x^2 + 4). Solution: The product, quotient, and reverse quotient of the given functions are as follows: Product (f g)(x): $$(f g)(x) = (4x^2 + x^4)\cdot\sqrt{x^2 + 4}$$ Quotient (f/g)(x): $$(f / g)(x) = \frac{4x^2 + x^4}{\sqrt{x^2 + 4}}$$ Reverse Quotient (g/f)(x): $$(g / f)(x) = \frac{\sqrt{x^2 + 4}}{4x^2 + x^4}$$

Step 1: 1. Find the product of the two functions (f g)(x)

To find the product of the two functions f(x) and g(x), we simply multiply the expressions for f(x) and g(x) together: $$(f g)(x) = f(x)g(x) = (4x^2 + x^4)\cdot\sqrt{x^2 + 4}$$

Step 2: 2. Simplify the product of the two functions (f g)(x)

The product of the two functions is already simplified in the initial form: $$(f g)(x) = (4x^2 + x^4)\cdot\sqrt{x^2 + 4}$$

Step 3: 3. Find the quotient of the two functions (f/g)(x)

To find the quotient of the two functions f(x) and g(x), we simply divide the expression for f(x) by the expression for g(x): $$(f / g)(x) = \frac{f(x)}{g(x)} = \frac{4x^2 + x^4}{\sqrt{x^2 + 4}}$$

Step 4: 4. Simplify the quotient of the two functions (f/g)(x)

The quotient of the two functions is already simplified in the initial form: $$(f / g)(x) = \frac{4x^2 + x^4}{\sqrt{x^2 + 4}}$$

Step 5: 5. Find the reverse quotient of the two functions (g/f)(x)

To find the reverse quotient of the two functions, i.e., the quotient of g(x) divided by f(x), simply invert the previous quotient: $$(g / f)(x) = \frac{g(x)}{f(x)} = \frac{\sqrt{x^2 + 4}}{4x^2 + x^4}$$

Step 6: 6. Simplify the reverse quotient of the two functions (g/f)(x)

The reverse quotient of the two functions is already simplified in the initial form: $$(g / f)(x) = \frac{\sqrt{x^2 + 4}}{4x^2 + x^4}$$ Thus, we have calculated the product, quotient, and reverse quotient of the given functions: $$(f g)(x) = (4x^2 + x^4)\cdot\sqrt{x^2 + 4}$$ $$(f / g)(x) = \frac{4x^2 + x^4}{\sqrt{x^2 + 4}}$$ $$(g / f)(x) = \frac{\sqrt{x^2 + 4}}{4x^2 + x^4}$$

Key Concepts

Function Multiplication

Function multiplication involves combining two functions by multiplying their outputs together. When you have two functions, say \( f(x) \) and \( g(x) \), you form the product function \( (f \cdot g)(x) \) by multiplying the expressions representing these functions.
For instance, if we have \( f(x) = 4x^2 + x^4 \) and \( g(x) = \sqrt{x^2 + 4} \), the multiplication is done as:

• Combine the functions: \((f \cdot g)(x) = f(x) \cdot g(x)\)
• Substitute their expressions: \((f \cdot g)(x) = (4x^2 + x^4) \cdot \sqrt{x^2 + 4}\)

At this stage, our product is in its simplest form, especially given the complexity of \( g(x) \). Notice how multiplication in algebra requires each term in one function to be multiplied by every term in the other function if both are polynomial expressions, but here one function is a polynomial and the other a square root, leaving us without further simplification. Using properties of exponents and roots, additional algebraic simplification may be possible if further required.

Function Division

Function division is the process of dividing one function by another. This is useful in finding ratios of function outputs and is represented as \((f / g)(x)\) or \((g / f)(x)\) depending on the arrangement.
To illustrate, consider the functions \( f(x) = 4x^2 + x^4 \) and \( g(x) = \sqrt{x^2 + 4} \):

• Form the quotient \((f / g)(x)\): It is \( \frac{f(x)}{g(x)} = \frac{4x^2 + x^4}{\sqrt{x^2 + 4}} \)
• For the reverse \((g / f)(x)\): It becomes \( \frac{g(x)}{f(x)} = \frac{\sqrt{x^2 + 4}}{4x^2 + x^4} \)

The key in division is knowing that you cannot divide by zero, hence it’s crucial to ascertain when the denominator is non-zero to ensure function division is valid. Small, rational simplifications may follow from canceling common factors, if possible, or finding co-factors, especially when dealing with fraction-based quotients.

Algebraic Simplification

Algebraic simplification is an essential part of managing function operations efficiently. It involves cleaning up expressions by reducing them into simpler or more compact forms without changing their underlying value.
Consider our previous functions and operations:

• Product: \((f \cdot g)(x) = (4x^2 + x^4) \cdot \sqrt{x^2 + 4}\)
• Quotients: \((f / g)(x) = \frac{4x^2 + x^4}{\sqrt{x^2 + 4}}\), \((g / f)(x) = \frac{\sqrt{x^2 + 4}}{4x^2 + x^4}\)

In each, simplifying begins by identifying like terms or factors that can be eliminated. For example, powers of \(x\) could be combined or reduced using exponent rules. Square roots can sometimes be rewritten using fractional exponents, assisting in algebraic manipulation. Although complex expressions might not simplify further, recognizing distributed components or simplifying common mathematical errors, such as mismanaging signs or terms, enhances clarity and presentation.