Question

Let \(f(x)=4 x^6\) and \(g(x)=2 x^3\). Find \(\left(\frac{f}{g}\right)(x)\). Then evaluate the quotient when \(x=5\).

Short answer

The quotient function results in \(2x^3\). When \(x=5\), the quotient function evaluates to \(2*(5)^3 = 250\).

Step-by-step solution

Function Quotient Rule

According to the quotient rule of functions, \(\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}\). Thus, \(\left(\frac{f}{g}\right)(x) = \frac{4 x^6}{2 x^3}\).

Simplification

The goal of this step is to simplify the expression from step 1. Looking at the expression, 4 in the numerator can be divided by 2 in the denominator, simplifying to 2. Similarly, the term \(x^6\) can be divided by \(x^3\). This is because when a term with an exponent in the numerator is divided by the same term with an exponent in the denominator, the resultant term is that term with the original exponents subtracted. In other words, if \(x^n\) is being divided by \(x^m\), the result is \(x^{n-m}\). Using this rule, \(x^6\) divided by \(x^3\) equals \(x^3\). This makes the simplified expression \(2x^3\).

Evaluating the Quotient

Once the simplified quotient function is achieved, the next step is to evaluate it when \(x=5\). Substituting \(x = 5\) into the simplified function is done as \(\left(\frac{f}{g}\right)(5) = 2*(5)^3\).

Key concepts

Quotient Rule for Functions

The quotient rule for functions is a powerful tool that allows us to divide one function by another. This rule is particularly useful when dealing with expressions where two functions, say \( f(x) \) and \( g(x) \), are involved, and you are asked to find their quotient.

Essentially, the rule states that if you have two functions, \( f(x) \) and \( g(x) \), the quotient \( \left(\frac{f}{g}\right)(x) \) is simply the division of these two functions:

• \( \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} \)

It is important to note that this rule is only applicable when \( g(x) eq 0 \), as division by zero is undefined.

In the given problem, we apply this rule by substituting \( f(x) = 4x^6 \) and \( g(x) = 2x^3 \) and dividing them to find \( \left(\frac{f}{g}\right)(x) \).

Polynomial Division

Polynomial division is a method used to simplify expressions where one polynomial is divided by another. For this process, we use the basic properties of exponents and coefficients.

In our example, after applying the quotient rule, we have the expression \( \frac{4x^6}{2x^3} \). To simplify this:

• First, divide the coefficients: 4 divided by 2 equals 2.
• Next, handle the exponents: \( x^6 \) divided by \( x^3 \) results in \( x^{6-3} = x^3 \).

Thus, the simplified form of the quotient is \( 2x^3 \). Polynomial division is all about breaking down each part of the expression separately to make it simpler.

Evaluating Functions

Evaluating functions involves substituting specific values into a function and calculating the result. Once a function is simplified, like our \( 2x^3 \) in the previous example, you can find its value for a specific \( x \) value by replacing \( x \) with that number.

For instance, to evaluate \( \left(\frac{f}{g}\right)(x) \) at \( x=5 \):

• Substitute 5 into the simplified function: \( 2(5)^3 \).
• Calculate the power: \( 5^3 = 125 \).
• Multiply by the coefficient: \( 2 \times 125 = 250 \).

So, \( \left(\frac{f}{g}\right)(5) = 250 \). Evaluating functions helps us determine the output of a function for any given input, making it a vital process in mathematics.