Given The Function G ( X ) = X + 3 X 2 + 2 X − 3 G(x) = \frac{x+3}{x^2 + 2x - 3} G ( X ) = X 2 + 2 X − 3 X + 3 , Determine The Nature Of The Function At Each Of The Following Values Of X X X . Indicate Whether G G G Has A Zero, A Vertical Asymptote, Or A Removable Discontinuity.Options:-

Mar 13, 2025 by ADMIN 291 views

**Analyzing the Nature of the Function g(x)**

Introduction

In this article, we will delve into the world of mathematical functions and explore the nature of the function $g(x) = \frac{x+3}{x^2 + 2x - 3}$ . We will examine the behavior of this function at specific values of $x$ and determine whether it has a zero, a vertical asymptote, or a removable discontinuity. This analysis will provide valuable insights into the properties of the function and its behavior.

Understanding the Function

The function $g(x)$ is a rational function, which means it is the ratio of two polynomials. The numerator of the function is $x+3$ , and the denominator is $x^2 + 2x - 3$ . To understand the nature of the function, we need to examine the behavior of the denominator, as it will determine the presence of any vertical asymptotes or removable discontinuities.

Factoring the Denominator

To analyze the behavior of the denominator, we need to factor it. The denominator can be factored as follows:

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

This factorization reveals that the denominator has two distinct factors: $(x + 3)$ and $(x - 1)$ . This information will be crucial in determining the nature of the function.

Analyzing the Function at Specific Values of x

Now that we have factored the denominator, we can analyze the function at specific values of $x$ . We will examine the behavior of the function at $x = -3$ , $x = 1$ , and $x = 0$ .

x = -3

At $x = -3$ , the denominator becomes $(x + 3)(x - 1) = (-3 + 3)(-3 - 1) = 0 \cdot (-4) = 0$ . Since the denominator is equal to zero, the function is undefined at $x = -3$ . However, we need to examine the numerator to determine whether the function has a zero or a removable discontinuity.

The numerator at $x = -3$ is $x + 3 = -3 + 3 = 0$ . Since both the numerator and denominator are equal to zero, the function has a removable discontinuity at $x = -3$ .

x = 1

At $x = 1$ , the denominator becomes $(x + 3)(x - 1) = (1 + 3)(1 - 1) = 4 \cdot 0 = 0$ . Since the denominator is equal to zero, the function is undefined at $x = 1$ . However, we need to examine the numerator to determine whether the function has a zero or a removable discontinuity.

The numerator at $x = 1$ is $x + 3 = 1 + 3 = 4$ . Since the numerator is not equal to zero, the function has a vertical asymptote at $x = 1$ .

x = 0

At $x = 0$ , the denominator becomes $(x + 3)(x - 1) = (0 + 3)(0 - 1) = 3 \cdot (-1) = -3$ . Since the denominator is not equal to zero, the function is defined at $x = 0$ . We need to examine the numerator to determine whether the function has a zero.

The numerator at $x = 0$ is $x + 3 = 0 + 3 = 3$ . Since the numerator is not equal to zero, the function does not have a zero at $x = 0$ .

Conclusion

In conclusion, we have analyzed the nature of the function $g(x) = \frac{x+3}{x^2 + 2x - 3}$ at specific values of $x$ . We have determined that the function has a removable discontinuity at $x = -3$ , a vertical asymptote at $x = 1$ , and is defined at $x = 0$ . This analysis has provided valuable insights into the properties of the function and its behavior.

Recommendations

Based on our analysis, we recommend the following:

To determine the nature of the function at other values of $x$ , we need to examine the behavior of the denominator and the numerator.
To remove the removable discontinuity at $x = -3$ , we need to factor the numerator and cancel out the common factor.
To analyze the behavior of the function at other values of $x$ , we need to use the factorization of the denominator to determine the presence of any vertical asymptotes or removable discontinuities.

Future Work

In future work, we plan to:

Analyze the behavior of the function at other values of $x$ .
Determine the nature of the function at other values of $x$ .
Investigate the properties of the function and its behavior.

References

[1] "Rational Functions" by Math Open Reference.
[2] "Discontinuities" by Wolfram MathWorld.

Appendix

The following is the code used to generate the plots:

import numpy as np
import matplotlib.pyplot as plt
def g(x):
return (x + 3) / (x**2 + 2*x - 3)

x = np.linspace(-10, 10, 400)

y = g(x)

plt.plot(x, y)
plt.title("Plot of g(x)")
plt.xlabel("x")
plt.ylabel("g(x)")
plt.grid(True)
plt.axhline(0, color='black')
plt.axvline(0, color='black')
plt.show()

Introduction

In our previous article, we analyzed the nature of the function $g(x) = \frac{x+3}{x^2 + 2x - 3}$ . We determined that the function has a removable discontinuity at $x = -3$ , a vertical asymptote at $x = 1$ , and is defined at $x = 0$ . In this article, we will answer some frequently asked questions about the function and its behavior.

Q: What is the nature of the function at x = -3?

A: The function has a removable discontinuity at $x = -3$ . This means that the function is undefined at $x = -3$ , but the discontinuity can be removed by factoring the numerator and canceling out the common factor.

Q: What is the nature of the function at x = 1?

A: The function has a vertical asymptote at $x = 1$ . This means that the function approaches positive or negative infinity as $x$ approaches 1.

Q: What is the nature of the function at x = 0?

A: The function is defined at $x = 0$ . This means that the function has a finite value at $x = 0$ .

Q: How do I determine the nature of the function at other values of x?

A: To determine the nature of the function at other values of $x$ , you need to examine the behavior of the denominator and the numerator. If the denominator is equal to zero, the function may have a vertical asymptote or a removable discontinuity. If the numerator is equal to zero, the function may have a zero.

Q: How do I remove the removable discontinuity at x = -3?

A: To remove the removable discontinuity at $x = -3$ , you need to factor the numerator and cancel out the common factor. In this case, the numerator is $x + 3$ , and the common factor is $(x + 3)$ . Canceling out the common factor, you get:

\frac{x+3}{x^2 + 2x - 3} = \frac{1}{x - 1}

This simplified function has a vertical asymptote at $x = 1$ .

Q: What are some common mistakes to avoid when analyzing the nature of a function?

A: Some common mistakes to avoid when analyzing the nature of a function include:

Not examining the behavior of the denominator and the numerator.
Not factoring the denominator to determine the presence of any vertical asymptotes or removable discontinuities.
Not canceling out common factors in the numerator and denominator.
Not using the correct notation and terminology.

Q: What are some real-world applications of analyzing the nature of a function?

A: Analyzing the nature of a function has many real-world applications, including:

Physics: Analyzing the behavior of physical systems, such as the motion of objects or the behavior of electrical circuits.
Engineering: Analyzing the behavior of complex systems, such as bridges or buildings.
Economics: Analyzing the behavior of economic systems, such as the behavior of stock prices or the behavior of interest rates.
Computer Science: Analyzing the behavior of algorithms and data structures.