Question

Find \((f g)(x)\) and \(\left(\frac{f}{g}\right)(x)\) and state the domain of each. Then evaluate \(f g\) and \(\frac{f}{g}\) for the given value of \(x\). $$ f(x)=x^4, g(x)=3 \sqrt{x} ; x=4 $$

Short answer

The function \((f g)(x) = 81x^2\) with domain x ≥ 0 evaluates to 1296 at x = 4. The function \(\left(\frac{f}{g}\right)(x) = \frac{1}{3}x^{7/2}\) with domain x > 0 also evaluates to 32 at x = 4.

Step-by-step solution

Define the given functions

The functions are defined as \(f(x)=x^4\) and \(g(x)=3 \sqrt{x}\).

Compute the composition of the functions

The composition of the functions is given by \((f g)(x) = f(g(x))\). By substituting \(g(x)\) into \(f(x)\), we get that \((f g)(x) = (3 \sqrt{x})^4 = 81x^2\).

Define the domain of the composition of functions

The domain of \((f g)(x)\) is the set of all real x for which this composition is defined. In this case, all real x, x ≥ 0.

Compute the division of the functions

The division of the functions is given by \(\frac{f}{g} = \frac{f(x)}{g(x)}\). By substituting \(f(x)\) and \(g(x)\) we get that \(\left(\frac{f}{g}\right)(x) = \frac{x^4}{3 \sqrt{x}} = \frac{1}{3}x^{7/2}\).

Define the domain of the division of functions

The domain of \(\left(\frac{f}{g}\right)(x)\) is the set of all real x for which this division is defined. Like the composition, it is all real x, x > 0 (not just ≥ 0 because we do not want to divide by 0).

Evaluate the functions for x = 4

Evaluate \((f g)(x)\) and \(\left(\frac{f}{g}\right)(x)\) for x = 4. We get \((f g)(4) = 81(4)^2 = 1296\) and \(\left(\frac{f}{g}\right)(4) = \frac{1}{3}(4)^{7/2} = 32\).

Key concepts

Domain of a Function

Understanding the domain of a function is crucial, as it tells us the set of all possible input values, typically referred to as "x", for which the function is defined. For example, consider a simple algebraic function like a square root, such as \( g(x) = 3 \sqrt{x} \). The square root function requires that \( x \geq 0 \) because you cannot take the square root of a negative number in the set of real numbers. Similarly, for a fraction like \( \frac{f}{g} \), we must ensure the denominator \( g(x) \) is never zero, as this would make the expression undefined.

For this particular exercise, when we defined the domain for the composition \((f g)(x) = 81x^2\) and the division \(\left(\frac{f}{g}\right)(x) = \frac{1}{3}x^{7/2}\):

• The domain of \((f g)(x)\) was determined to be all real \(x\) where \(x \geq 0\) because it can include zero.
• In contrast, the domain of \(\left(\frac{f}{g}\right)(x)\) is \(x > 0\) to avoid division by zero.

These specifications help prevent any mathematical inconsistencies and ensure the functions are well-defined over their respective domains.

Composite Functions

Composite functions involve combining two functions to create a new one, where you substitute one function into another. For example, given two functions \( f(x) \) and \( g(x) \), the composite function \((f g)(x)\) is found by substituting \( g(x) \) into \( f(x) \).

This technique can extend the power of basic functions, creating more complex expressions. In our exercise, the calculation of \((f g)(x) = f(g(x))\) resulted in substituting \(g(x) = 3 \sqrt{x}\) into \(f(x) = x^4\), simplifying to:

• \((f g)(x) = (3 \sqrt{x})^4 = 81x^2\)

This result shows how combining two functions can significantly alter the output expression, expanding its complexity and application.

Composite functions are often used to model complex real-world scenarios where multiple processes occur sequentially. Therefore, understanding how to construct and manipulate them is invaluable in both mathematics and applied sciences.

Division of Functions

Like composite functions, division of functions involves operating with two functions. Here, you divide one function by another, written as \(\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}\). It's crucial to consider the function's domain so the divisor \(g(x)\) is never zero, which would otherwise make the expression undefined.

In this exercise, dividing \(f(x) = x^4\) by \(g(x) = 3 \sqrt{x}\), we calculate:

• \(\left(\frac{f}{g}\right)(x) = \frac{x^4}{3 \sqrt{x}} = \frac{1}{3}x^{7/2}\)

Ensuring the domain excludes \(x = 0\) is essential here. This provision prevents errors and maintains the integrity of the function's expression.

Division of functions is beneficial in many analytical applications, allowing ratios and comparisons between varying factors in a formulaic context. Understanding this concept empowers one to approach complex mathematical issues with confidence and precision. Ultimately, mastering function operations like these lays the groundwork for tackling even more advanced topics in calculus and algebra.