Problem 79 - Algebra 2

Question

Let \(f(x)=3 x-2\) and \(g(x)=x^{2}+1 .\) Perform each function operation and use the properties of real numbers to justify each step in simplifying your answer. $$ \frac{g(x)}{f(x)} $$

Step-by-Step Solution

Answer

\(\frac{g(x)}{f(x)} = \frac{x^2 + 1}{3x - 2}\)

Step 1: Write Down Both Functions

Let’s start by writing down both functions. So we have \(f(x)=3x-2\) and \(g(x)=x^2+1\).

Step 2: Perform the Division

We’re required to perform \(\frac{g(x)}{f(x)}\), which would be \(\frac{x^2 + 1}{3x - 2}\), since \(g(x) = x^2 + 1\) and \(f(x) = 3x - 2\).

Step 3: Simplify the Expression

There is no algebraic simplification to be done in this case. The function \(\frac{x^2 + 1}{3x - 2}\) cannot be simplified using the properties of real numbers in this scenario. So, \(\frac{g(x)}{f(x)} = \frac{x^2 + 1}{3x - 2}\).

Key Concepts

Division of Functions

Dividing functions is similar to dividing numbers, except instead of working with plain numbers, we are working with function expressions. In mathematics, when we perform division with functions, we are finding a new function that represents the ratio of the two given functions.
To divide the functions, we simply write the dividend function (numerator) over the divisor function (denominator) in a fraction form. For our example, we have the functions given by:

• The dividend function, \(g(x) = x^2 + 1\)
• The divisor function, \(f(x) = 3x - 2\)

By dividing these, we get \(\frac{g(x)}{f(x)} = \frac{x^2 + 1}{3x - 2}\).
This result is a new function which will give a value when you plug in a value of \(x\) into both \(g(x)\) and \(f(x)\), provided that \(f(x) eq 0\), since division by zero is undefined.

Properties of Real Numbers

The properties of real numbers are essential truths about numbers that help us perform algebraic operations more efficiently and correctly. When dealing with functions, these properties play a key role in combining and simplifying expressions.
Here are some important properties that can be used during function operations:

• Commutative Property: For addition and multiplication, the order in which you add or multiply numbers does not change their sum or product.
• Associative Property: For addition and multiplication, the way numbers are grouped does not change their sum or product.
• Distributive Property: Multiplying a number by a group of numbers added together is the same as doing each multiplication separately.

In the division of functions, these properties don't directly change the essence of the operation but understanding them ensures any further operations on the resulting function are handled correctly.

Simplifying Expressions

Simplifying an expression involves reducing it to its most basic form without altering its value. The aim is to make expressions easier to understand and work with.
For some expressions, simplifying might involve combining like terms, factoring, or canceling out terms. However, in function operations like division, simplification might not always be possible, as seen in our example.
In the example of dividing \(g(x)\) by \(f(x)\), the fraction \(\frac{x^2 + 1}{3x - 2}\) cannot be simplified further, as there are no common factors between the numerator and the denominator.
Remember, failing to simplify an expression further doesn’t mean the work is incorrect—it simply means it's already in its simplest possible form given the operations and values involved.