Select The Correct Answer.Consider Functions { F$}$ And { G$} . . . { F(x)=\frac{x^2-16}{x^2+3x-10}, \quad \text{for } X \neq -5 \text{ And } X \neq 2 \} $[ G(x)=\frac{x 2-4}{x 2-7x+12}, \quad \text{for } X \neq 3 \text{

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Simplifying Rational Functions: A Comparison of f(x) and g(x)

Introduction

When dealing with rational functions, simplifying them is crucial to understand their behavior and properties. In this article, we will explore two rational functions, f(x) and g(x), and compare their simplified forms. By analyzing these functions, we will gain a deeper understanding of how to simplify rational expressions and identify key characteristics of these functions.

Function f(x)

The first rational function is given by:

f(x)=x2−16x2+3x−10,for x≠−5 and x≠2{ f(x)=\frac{x^2-16}{x^2+3x-10}, \quad \text{for } x \neq -5 \text{ and } x \neq 2 }

To simplify this function, we need to factor both the numerator and the denominator.

import sympy as sp

x = sp.symbols('x')



f = (x2 - 16) / (x2 + 3*x - 10)



numerator_factors = sp.factor(x2 - 16)
denominator_factors = sp.factor(x2 + 3*x - 10)


print("Numerator factors:", numerator_factors)
print("Denominator factors:", denominator_factors)

By factoring the numerator and denominator, we get:

f(x)=(x+4)(x−4)(x+5)(x−2){ f(x)=\frac{(x+4)(x-4)}{(x+5)(x-2)} }

Function g(x)

The second rational function is given by:

g(x)=x2−4x2−7x+12,for x≠3{ g(x)=\frac{x^2-4}{x^2-7x+12}, \quad \text{for } x \neq 3 }

To simplify this function, we also need to factor both the numerator and the denominator.

# Define the function g(x)
g = (x**2 - 4) / (x**2 - 7*x + 12)


numerator_factors = sp.factor(x2 - 4)
denominator_factors = sp.factor(x2 - 7*x + 12)


print("Numerator factors:", numerator_factors)
print("Denominator factors:", denominator_factors)

By factoring the numerator and denominator, we get:

g(x)=(x+2)(x−2)(x−3)(x−4){ g(x)=\frac{(x+2)(x-2)}{(x-3)(x-4)} }

Comparison of f(x) and g(x)

Now that we have simplified both functions, let's compare their forms.

f(x)=(x+4)(x−4)(x+5)(x−2){ f(x)=\frac{(x+4)(x-4)}{(x+5)(x-2)} }

g(x)=(x+2)(x−2)(x−3)(x−4){ g(x)=\frac{(x+2)(x-2)}{(x-3)(x-4)} }

We can see that both functions have a similar structure, with a numerator that is a product of two binomials and a denominator that is also a product of two binomials. However, the coefficients and the signs of the terms in the numerator and denominator are different.

Key Takeaways

In this article, we have simplified two rational functions, f(x) and g(x), and compared their forms. We have seen that both functions have a similar structure, but with different coefficients and signs. This comparison has helped us understand the properties of rational functions and how to simplify them.

Conclusion

Simplifying rational functions is an essential step in understanding their behavior and properties. By comparing the simplified forms of f(x) and g(x), we have gained a deeper understanding of how to simplify rational expressions and identify key characteristics of these functions. This knowledge can be applied to a wide range of mathematical problems and is an essential tool for any mathematician or scientist.

Future Work

In the future, we can explore more complex rational functions and compare their simplified forms. We can also investigate the properties of these functions, such as their domain and range, and how they behave under different transformations.

References

Introduction

In our previous article, we explored the simplification of rational functions and compared the forms of two rational functions, f(x) and g(x). In this article, we will answer some common questions related to simplifying rational functions and provide additional insights into this topic.

Q: What is a rational function?

A: A rational function is a function that can be expressed as the ratio of two polynomials. In other words, it is a function of the form:

f(x)=p(x)q(x){ f(x)=\frac{p(x)}{q(x)} }

where p(x) and q(x) are polynomials.

Q: How do I simplify a rational function?

A: To simplify a rational function, you need to factor both the numerator and the denominator. This involves finding the greatest common factor (GCF) of the numerator and the denominator and canceling out any common factors.

Q: What is the greatest common factor (GCF)?

A: The greatest common factor (GCF) of two polynomials is the largest polynomial that divides both of them evenly. For example, the GCF of x^2 + 3x and x^2 + 5x is x.

Q: How do I find the GCF of two polynomials?

A: To find the GCF of two polynomials, you can use the following steps:

  1. Factor both polynomials.
  2. Identify the common factors.
  3. Multiply the common factors together to get the GCF.

Q: What is the difference between a rational function and a rational expression?

A: A rational function is a function that can be expressed as the ratio of two polynomials, while a rational expression is a single expression that is the ratio of two polynomials. In other words, a rational function is a function that can be expressed as:

f(x)=p(x)q(x){ f(x)=\frac{p(x)}{q(x)} }

while a rational expression is simply:

p(x)q(x){ \frac{p(x)}{q(x)} }

Q: Can I simplify a rational expression?

A: Yes, you can simplify a rational expression by factoring both the numerator and the denominator and canceling out any common factors.

Q: What are some common mistakes to avoid when simplifying rational functions?

A: Some common mistakes to avoid when simplifying rational functions include:

  • Not factoring the numerator and denominator properly.
  • Not canceling out common factors correctly.
  • Not checking for any remaining factors in the numerator or denominator.

Q: How do I check if a rational function is simplified?

A: To check if a rational function is simplified, you can follow these steps:

  1. Factor both the numerator and the denominator.
  2. Cancel out any common factors.
  3. Check if there are any remaining factors in the numerator or denominator.

Q: Can I use technology to simplify rational functions?

A: Yes, you can use technology such as calculators or computer software to simplify rational functions. However, it's always a good idea to double-check your work by factoring and canceling out common factors manually.

Conclusion

Simplifying rational functions is an essential step in understanding their behavior and properties. By following the steps outlined in this article, you can simplify rational functions and gain a deeper understanding of this topic. Remember to always check for any remaining factors in the numerator or denominator and to use technology to simplify rational functions when necessary.

References