Problem 11 - Algebra 2

Question

Let \(f(x)=3 x+5\) and \(g(x)=x^{2} .\) Perform each function operation. $$ \left(\frac{f}{g}\right)(x) $$

Step-by-Step Solution

Answer

\(\frac{f}{g}(x) = \frac{3x+5}{x^2}\)

Step 1: Setting up the functions

Firstly, the two given functions, \(f(x)\) and \(g(x)\), are identified as: \(f(x) = 3x + 5\) and \(g(x) = x^2\).

Step 2: Substituting functions into the operation

Substitute \(\frac{f}{g}(x)\) with the given definitions of \(f(x)\) and \(g(x)\), you obtain: \(\frac{f}{g}(x) = \frac{3x+5}{x^2}\).

Step 3: Dividing the functions

Proceed to simplify \(\frac{3x+5}{x^2}\). It's already in simplest form, as there's no common factor between the numerator and the denominator that could be divided out.

Key Concepts

Function Division

In mathematics, dividing functions is a way to combine two functions into one by representing one function as the numerator and the other as the denominator. For example, if you have two functions, \(f(x)\) and \(g(x)\), the function division \(\frac{f}{g}(x)\) is the quotient of \(f(x)\) and \(g(x)\). Function division is usually used to create a new function, which is then evaluated for a domain of values where \(g(x) eq 0\).
To divide functions in general, you might follow these steps:

• Identify the given functions.
• Rewrite the division operation by placing the functions into a single fraction.
• Check if the result can be simplified by factoring or canceling common factors.

As seen here, was that the resulting expression \(\frac{3x+5}{x^2}\) which is obtained from substituting into \(\frac{f}{g}(x)\).
Remember, division by zero is undefined, so always ensure \(g(x) eq 0\).

Rational Functions

Rational functions are expressions formed by dividing one polynomial by another polynomial. Much like rational numbers, which are ratios of integers, rational functions are ratios of polynomials. A rational function is typically written in the form \(\frac{p(x)}{q(x)}\), where both \(p(x)\) and \(q(x)\) are polynomials, and \(q(x) eq 0\).

The function \(\frac{3x+5}{x^2}\) from the exercise is a rational function. Here, we have a linear function in the numerator \(3x+5\) and a quadratic function \(x^2\) in the denominator. Rational functions are usually analyzed for their behavior at values of \(x\) where the denominator becomes zero, as these points are either holes or vertical asymptotes in the graph of the function. Always assess the function's domain to exclude these values.
In our example, \(x^2 eq 0\), so \(x eq 0\) is part of the domain condition.

Simplifying Expressions

Simplifying expressions often involves reducing fractions to their simplest form by canceling out any common factors from the numerator and the denominator. Simplification rules:

• Factor polynomials if possible.
• If a factor is common to both the numerator and the denominator, cancel them.
• Check for possibilities of further simplification like common terms.

However, in our given expression \(\frac{3x+5}{x^2}\), the task doesn't present any common factors between the numerator \(3x+5\) and the denominator \(x^2\).
Thus, it remains as it is with no further simplification required. It is vital to always attempt simplification for ease of further operations and clear analysis, but this expression is already in its simplest form.

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. The basic format of a polynomial function in terms of \(x\) is \(a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0\), where \(n\) is a non-negative integer, and the coefficients \(a_i\) are real numbers.

In the provided exercise, \(f(x)=3x+5\) is a polynomial function of degree 1 (linear polynomial), and \(g(x)=x^2\) is a polynomial function of degree 2 (quadratic polynomial). These polynomials are the fundamental elements within our rational function \(\frac{3x+5}{x^2}\).
A solid understanding of polynomial functions is essential for analyzing rational functions because their roots, degrees, and behavior can influence the overall function, especially when evaluated against various operations such as division.