Problem 93 - Algebra 2

Question

Let \(g(x)=x^{2}-4\) and \(h(x)=4 x-6 .\) Find \(\left(\frac{g}{h}\right)(x)\)

Step-by-Step Solution

Answer

The result of \(\left(\frac{g}{h}\right)(x)\) is \(\frac{x^{2}-4}{4x-6}\).

Step 1: Write Down The Expressions

Firstly, write down the expressions you're given: \(g(x)=x^{2}-4\) and \(h(x)=4 x-6\).

Step 2: Perform The Division

Next, divide \(g(x)\) by \(h(x)\) to find \(\left(\frac{g}{h}\right)(x)\). Remember to divide each term in the numerator by the whole denominator.

Step 3: Simplify

Simplify the obtained fraction. This assures the final result is as simple and straightforward as possible.

Key Concepts

Division of Functions

When dealing with the division of functions, we begin by identifying two functions that need to be divided. In the exercise at hand, the functions are given as \(g(x) = x^{2} - 4\) and \(h(x) = 4x - 6\).
To perform the division of these functions, we form a new function \(\left(\frac{g}{h}\right)(x)\). This involves dividing the expression for \(g(x)\) by the expression for \(h(x)\), meaning our fraction becomes:\[\left(\frac{g}{h}\right)(x) = \frac{x^{2} - 4}{4x - 6}\]
When we divide functions, the result is also a function, and it is important to state that the denominator, \(h(x)\), should never be zero. If \(h(x)\) equals zero, the function division is undefined because division by zero is not possible in mathematics. Therefore, it's good practice to determine where \(h(x) = 0\) and restrict the function division accordingly.
In our case, solving \(4x - 6 = 0\) gives the restriction that \(x eq \frac{3}{2}\).

Simplifying Expressions

After performing the division of functions, often the resulting expression can be simplified to make it more manageable. Simplifying expressions involves performing operations that reduce them to their most basic form. The main goal is to make the expression easier to work with or understand.
In our example, the fraction obtained from dividing the two functions needs to be simplified. The given expression is \(\frac{x^{2} - 4}{4x - 6}\). A key step in simplifying is to factor both the numerator and the denominator whenever possible.
For the numerator, \(x^{2} - 4\), we recognize it as a difference of squares, which factors to \((x-2)(x+2)\). The denominator \(4x-6\) can be factored by taking out the greatest common factor, giving \(2(2x-3)\). Therefore, we have:\[\frac{(x-2)(x+2)}{2(2x-3)}\]In this case, no terms are common between the numerator and the denominator that allow for cancellation. It's as simple as this expression gets based on these terms. Simplification ensures the expression is clear and practical for further operations.

Function Operations

Function operations are methods that extend basic arithmetic to functions. These operations include addition, subtraction, multiplication, and division of functions. Each operation combines two functions to create a new one.
In our exercise, we focused on the division operation, which necessitated forming a quotient of the expressions from two given functions, \(g(x)\) and \(h(x)\). While performing these operations, it's important to pay special attention to the domain of the resulting function. The domain of \(\left(\frac{g}{h}\right)(x)\) will be influenced by the restrictions of both original functions, and in particular, it must exclude values that make the denominator zero.
Each operation impacts the properties and behaviors of functions differently. For instance:

• Adding functions results in a new function whose output is the sum of the outputs of the two functions at any given input.
• Subtracting functions does the same but results in the difference.
• Multiplying functions joins their outputs by multiplication.
• Dividing, like in this problem, involves forming a fraction.

Understanding these operations helps in tackling more intricate problems and facilitates the exploration of function behavior, especially in calculus and algebra.