Problem 6 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The result of the operation \(\frac{g(x)}{f(x)}\) with the given functions is \(\frac{x^2}{3x+5}\).

Step 1: Identify the Given Functions

The functions given are \(f(x) = 3x + 5\) and \(g(x) = x^2\). The operation being asked is \(\frac{g(x)}{f(x)}\).

Step 2: Substitute the Functions

Now substitute the given functions into the operation. Instead of \(\frac{g(x)}{f(x)}\), write it as \(\frac{x^2}{3x+5}\).

Step 3: Check for Simplification

Look at the resultant function and check if there is any further possibility for simplification. In this equation, \(\frac{x^2}{3x+5}\), no further simplification is possible.

Key Concepts

Algebraic Functions

Algebraic functions are a fundamental concept in mathematics. They are functions created using algebraic operations such as addition, subtraction, multiplication, division, and taking roots. These functions are expressed in the form of equations, where the variables can have real or complex values.
For example, the functions given in our problem are algebraic:

• Function \( f(x) = 3x + 5 \) is a linear algebraic function. It simply involves addition and multiplication.
• Function \( g(x) = x^2 \) is a polynomial algebraic function. It involves powers of the variable.

Algebraic functions are crucial as they form the building blocks for more complex expressions and help describe a wide range of real-world phenomena.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are both polynomials. In other words, they are ratios of algebraic expressions. Our task in the exercise was to perform a division operation that's a typical operation involving rational expressions.
In the problem, we have:

• Numerator: \( g(x) = x^2 \)
• Denominator: \( f(x) = 3x + 5 \)
• Formation: Resulting expression is \( \frac{x^2}{3x+5} \)

Understanding rational expressions involves knowing how to manipulate these equations properly and recognizing whether they can be simplified. If a simplification is possible, it often makes the expression easier to work with. However, in this case, no further simplification was possible.

Simplification of Expressions

Simplifying expressions plays a significant role in mathematics. It involves condensing an equation to its simplest form, making it less complicated and easier to analyze or compute. This process can involve combining like terms, reducing fractions, eliminating radicals, and factoring polynomials.
In our exercise, we assessed the expression \( \frac{x^2}{3x+5} \) for simplification:

• Checking common factors: Look for common factors in the numerator and the denominator. If there are none, as with our current expression, then no reduction is possible.
• Ensure the expression is valid for given variable constraints, like ensuring the denominator is not zero (as \( 3x + 5 eq 0 \)).

The simplification process not only affects the mathematical outcome but also how easily and efficiently a problem can be solved.