Problem 12 - Algebra 2

Question

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

Step-by-Step Solution

Answer

\((g/f)(x) = (x^2)/(3x + 5)\)

Step 1: Identify the Functions

The given functions are \(f(x) = 3x + 5\) and \(g(x) = x^2\).

Step 2: Perform the Function Division

Divide the two functions as shown: \((g/f)(x) = g(x)/f(x)\). By plugging in the functions for the variables we get \((g/f)(x) = (x^2)/(3x + 5)\).

Step 3: Simplify the Result

The division of the two functions cannot be further simplified. So, we leave it as it is: \((g/f)(x) = (x^2)/(3x + 5)\).

Key Concepts

Polynomial Functions

Polynomial functions are an essential component of algebra that can be characterized by expressions involving powers of variables. These functions take the form \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\), where \(a_n, a_{n-1}, \ldots, a_0\) are coefficients and \(n\) represents the highest power. Polynomial functions can have multiple terms, but their operations are quite straightforward due to their simplicity in being either added, subtracted, multiplied, or divided, with the functions retaining polynomial properties as long as division by zero is avoided. In our original problem, the polynomial function \(g(x) = x^2\) is utilized. It is a basic polynomial with just one term and exhibits a quadratic behavior. This kind of function is fundamental in many areas of mathematics, providing intuitive insights into more complex structures. For instance, quadratic functions are often used to model projectile motion or economic models as they illustrate parabolic trends.

Rational Expressions

Rational expressions are formed by dividing one polynomial by another. Essentially, any expression that can be expressed as \(\frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x) eq 0\), is a rational expression. Such expressions must be carefully handled, especially in terms of simplifying or factoring the numerator and denominator to avoid division by zero, which is undefined. In the exercise, \(\frac{g(x)}{f(x)}\) transforms into a rational expression, \(\frac{x^2}{3x + 5}\). This piece cannot be simplified further without specific values that may cancel terms in both the numerator and the denominator. Therefore, understanding how to manipulate and evaluate rational expressions, especially in applied scenarios, is crucial to mastering them.

Function Division

Function division involves finding the quotient of two functions. It is represented by \(\left(\frac{g}{f}\right)(x)\) and results in a new function created from dividing one original function by another. To carry out function division, you merely place the second function as the divisor under the first one. This process involves substituting the expressions for \(f(x)\) and \(g(x)\) and then ensuring the denominator does not equate to zero to avoid undefined expressions. In the original problem, the division yields \(\left(\frac{x^2}{3x + 5}\right)(x)\). There is a noteworthy point to keep in mind: even though the expression cannot be simplified further algebraically, its domain does not include values where the denominator becomes zero, meaning \(x eq -\frac{5}{3}\). This provides a clearer picture for when solving such exercises in practice.