Simplify The Quotient Of Two Functions F(x)/g(x)

Nov 3, 2025 by Admin 49 views

Hey guys! In this article, we're going to tackle a fun problem in mathematics: finding and simplifying the quotient of two functions. Specifically, we're given two functions, $f(x) = 1 - \frac{2}{x+2}$ and $g(x) = 1 + \frac{3}{x-2}$ , and our mission, should we choose to accept it, is to find $\frac{f(x)}{g(x)}$ and simplify it as much as possible. So, grab your pencils, and let's dive in!

Understanding the Functions

Before we jump into the division, let's take a closer look at our functions. Understanding their individual components will help us simplify the quotient more effectively. Our functions are $f(x) = 1 - \frac{2}{x+2}$ and $g(x) = 1 + \frac{3}{x-2}$ . The function f(x) involves a constant term subtracted by a rational expression, while g(x) involves a constant term added to a rational expression. To effectively handle these, we will combine terms to express them as single rational expressions. This involves finding common denominators and combining the numerators. It's like making sure everyone speaks the same language before a big meeting!

First, let’s simplify $f(x)$ . We can rewrite $1$ as $\frac{x+2}{x+2}$ , so we have:

$f(x) = \frac{x+2}{x+2} - \frac{2}{x+2}$

Now, we can combine the numerators since they have a common denominator:

$f(x) = \frac{x+2-2}{x+2} = \frac{x}{x+2}$

Next, let’s simplify $g(x)$ . We can rewrite $1$ as $\frac{x-2}{x-2}$ , so we have:

$g(x) = \frac{x-2}{x-2} + \frac{3}{x-2}$

Combine the numerators:

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

Now that we've simplified $f(x)$ and $g(x)$ individually, we're ready to tackle their quotient. Remember, simplifying each function separately makes the division process much smoother and less prone to errors. Think of it as prepping your ingredients before you start cooking – it just makes everything easier!

Dividing the Functions

Now that we have our simplified functions, $f(x) = \frac{x}{x+2}$ and $g(x) = \frac{x+1}{x-2}$ , we can find $\frac{f(x)}{g(x)}$ . Remember that dividing by a fraction is the same as multiplying by its reciprocal. So, we have:

$\frac{f(x)}{g(x)} = \frac{\frac{x}{x+2}}{\frac{x+1}{x-2}} = \frac{x}{x+2} \cdot \frac{x-2}{x+1}$

This simplifies to:

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

At this point, we have successfully divided $f(x)$ by $g(x)$ . However, to fully simplify, we should check if there are any common factors in the numerator and denominator that can be canceled out. This step ensures that our final answer is in its most reduced form. In this case, there are no common factors, so we can leave the expression as is. Factoring and canceling common terms is a critical skill in simplifying rational expressions, so always keep an eye out for opportunities to apply it!

Final Simplified Form

After performing the division and simplifying, we arrive at the final form of the quotient:

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

This expression is now in its simplest form. There are no common factors between the numerator and the denominator, and we have successfully performed the division as requested. To ensure the expression is fully simplified, we inspect the numerator and denominator to verify the absence of any common factors. When none are found, we know that the expression is in its most reduced form, completing our simplification process.

Therefore, the final answer is:

$\frac{x(x-2)}{(x+2)(x+1)}$

Expanding the Expression (Optional)

While the above form is perfectly acceptable and simplified, sometimes it's useful to expand the expression to see it in polynomial form. This isn't strictly necessary for simplification, but it can be helpful for further analysis or comparison with other expressions. Expanding the numerator, we get:

$x(x-2) = x^2 - 2x$

Expanding the denominator, we get:

$(x+2)(x+1) = x^2 + x + 2x + 2 = x^2 + 3x + 2$

So, the expanded form of the quotient is:

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

Both forms, the factored form $\frac{x(x-2)}{(x+2)(x+1)}$ and the expanded form $\frac{x^2 - 2x}{x^2 + 3x + 2}$ , are equivalent and represent the simplified quotient of the given functions. The choice of which form to use often depends on the context of the problem or the specific requirements of the situation.

Important Considerations

When working with rational functions, it's crucial to consider the domain. The domain consists of all possible input values (x-values) for which the function is defined. In the context of rational functions, we need to exclude any values that would make the denominator equal to zero, as division by zero is undefined.

For $f(x) = \frac{x}{x+2}$ , the denominator is $x+2$ . Setting this equal to zero gives $x = -2$ . Thus, $x$ cannot be $-2$ .

For $g(x) = \frac{x+1}{x-2}$ , the denominator is $x-2$ . Setting this equal to zero gives $x = 2$ . Thus, $x$ cannot be $2$ .

For the quotient $\frac{f(x)}{g(x)} = \frac{x(x-2)}{(x+2)(x+1)}$ , we also need to consider the values that make the original denominator $g(x)$ equal to zero, which is $x=2$ . Additionally, we must consider the values that make the denominator of the simplified expression equal to zero, which are $x = -2$ and $x = -1$ (from $x+1$ ).

Therefore, the domain of $\frac{f(x)}{g(x)}$ is all real numbers except $x = -2$ , $x = -1$ , and $x = 2$ . This ensures that the function is well-defined and avoids any undefined expressions.

Practical Applications

Understanding how to simplify the quotient of functions isn't just an abstract mathematical exercise. It has practical applications in various fields, including:

Engineering: In control systems, engineers often deal with transfer functions, which are ratios of polynomials. Simplifying these functions helps in analyzing the stability and performance of the system.
Physics: In electromagnetism and quantum mechanics, ratios of functions appear frequently. Simplifying these ratios can lead to easier calculations and clearer interpretations.
Economics: Economists use ratios of functions in modeling supply and demand curves. Simplifying these ratios can help in understanding market equilibrium and predicting market behavior.
Computer Science: In algorithm analysis, ratios of functions are used to describe the time complexity of algorithms. Simplifying these ratios can help in comparing the efficiency of different algorithms.

By mastering these simplification techniques, you're equipping yourself with a valuable tool that can be applied in a wide range of real-world scenarios. Keep practicing, and you'll become a pro in no time!

Conclusion

Alright, guys, that's a wrap! We've successfully found and simplified $\frac{f(x)}{g(x)}$ for the given functions $f(x)$ and $g(x)$ . Remember, the key steps were simplifying each function individually, dividing by multiplying by the reciprocal, and then looking for any common factors to cancel out. By following these steps, you can tackle similar problems with confidence. Keep practicing, and you'll become a master of function manipulation in no time! You rock!