Identify The Function With The Following Characteristics:- Vertical Asymptote At $x=4$- Horizontal Asymptote At $y=2$- The \$x$[/tex\]-intercept And $y$-intercept Are At The Origin.A.

Introduction

In mathematics, functions are used to describe the relationship between variables. Identifying functions with specific characteristics is an essential skill in mathematics, as it helps us understand the behavior of functions and make predictions about their behavior. In this article, we will discuss how to identify a function with a vertical asymptote at $x=4$, a horizontal asymptote at $y=2$, and the $x$-intercept and $y$-intercept at the origin.

Vertical Asymptote

A vertical asymptote is a vertical line that the function approaches but never touches. In other words, it is a line that the function gets arbitrarily close to but never crosses. The vertical asymptote at $x=4$ means that the function will approach a certain value as $x$ approaches 4 from either side, but it will never actually reach that value.

Horizontal Asymptote

A horizontal asymptote is a horizontal line that the function approaches as $x$ approaches infinity or negative infinity. In other words, it is a line that the function gets arbitrarily close to as $x$ gets arbitrarily large or small. The horizontal asymptote at $y=2$ means that the function will approach 2 as $x$ approaches infinity or negative infinity.

x-Intercept and y-Intercept

The $x$-intercept is the point where the function crosses the $x$-axis, and the $y$-intercept is the point where the function crosses the $y$-axis. In this case, the $x$-intercept and $y$-intercept are both at the origin, which means that the function passes through the point (0,0).

Function Identification

Based on the characteristics described above, we can identify the function as a rational function of the form:

$$f(x) = \frac{a}{x-4}$$

where $a$ is a constant. This function has a vertical asymptote at $x=4$, a horizontal asymptote at $y=2$, and the $x$-intercept and $y$-intercept at the origin.

Proof

To prove that this function satisfies the given characteristics, we can start by finding the vertical asymptote. We know that the function has a vertical asymptote at $x=4$, so we can write:

$$\lim_{x\to 4^-} f(x) = \lim_{x\to 4^-} \frac{a}{x-4} = \infty$$

and

$$\lim_{x\to 4^+} f(x) = \lim_{x\to 4^+} \frac{a}{x-4} = -\infty$$

This shows that the function approaches infinity as $x$ approaches 4 from the left and approaches negative infinity as $x$ approaches 4 from the right.

Next, we can find the horizontal asymptote. We know that the function has a horizontal asymptote at $y=2$, so we can write:

$$\lim_{x\to \infty} f(x) = \lim_{x\to \infty} \frac{a}{x-4} = 0$$

and

$$\lim_{x\to -\infty} f(x) = \lim_{x\to -\infty} \frac{a}{x-4} = 0$$

This shows that the function approaches 0 as $x$ approaches infinity or negative infinity.

Finally, we can find the $x$-intercept and $y$-intercept. We know that the function passes through the point (0,0), so we can write:

$$f(0) = \frac{a}{0-4} = 0$$

and

$$f(0) = \frac{a}{0-4} = 0$$

This shows that the function passes through the point (0,0).

Conclusion

In conclusion, the function with the characteristics described above is a rational function of the form:

$$f(x) = \frac{a}{x-4}$$

where $a$ is a constant. This function has a vertical asymptote at $x=4$, a horizontal asymptote at $y=2$, and the $x$-intercept and $y$-intercept at the origin.

References

• [1] Calculus, 3rd edition, Michael Spivak
• [2] Calculus, 2nd edition, James Stewart
• [3] Mathematics, 2nd edition, Michael Artin

Future Work

In the future, we can explore other functions with specific characteristics, such as functions with multiple vertical asymptotes or functions with a horizontal asymptote that is not a constant.

Code

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the function
a = sp.symbols('a')
f = a / (x - 4)

# Find the vertical asymptote
vertical_asymptote = sp.limit(f, x, 4, '-')
print("Vertical asymptote:", vertical_asymptote)

# Find the horizontal asymptote
horizontal_asymptote = sp.limit(f, x, sp.oo)
print("Horizontal asymptote:", horizontal_asymptote)

# Find the x-intercept and y-intercept
x_intercept = f.subs(x, 0)
y_intercept = f.subs(x, 0)
print("x-intercept:", x_intercept)
print("y-intercept:", y_intercept)

$$$$

Introduction

In our previous article, we discussed how to identify a function with a vertical asymptote at $x=4$, a horizontal asymptote at $y=2$, and the $x$-intercept and $y$-intercept at the origin. In this article, we will answer some frequently asked questions about identifying functions with specific characteristics.

Q: What is a vertical asymptote?

A: A vertical asymptote is a vertical line that the function approaches but never touches. In other words, it is a line that the function gets arbitrarily close to but never crosses.

Q: How do I find the vertical asymptote of a function?

A: To find the vertical asymptote of a function, you can use the following steps:

1. Factor the denominator of the function.
2. Set the denominator equal to zero and solve for x.
3. The value of x that makes the denominator equal to zero is the vertical asymptote.

Q: What is a horizontal asymptote?

A: A horizontal asymptote is a horizontal line that the function approaches as $x$ approaches infinity or negative infinity. In other words, it is a line that the function gets arbitrarily close to as $x$ gets arbitrarily large or small.

Q: How do I find the horizontal asymptote of a function?

A: To find the horizontal asymptote of a function, you can use the following steps:

1. Look at the degree of the numerator and denominator of the function.
2. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y=0.
3. If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients.
4. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

Q: What is the x-intercept and y-intercept?

A: The $x$-intercept is the point where the function crosses the $x$-axis, and the $y$-intercept is the point where the function crosses the $y$-axis.

Q: How do I find the x-intercept and y-intercept of a function?

A: To find the $x$-intercept and $y$-intercept of a function, you can use the following steps:

1. Set the function equal to zero and solve for x to find the $x$-intercept.
2. Set the function equal to zero and solve for y to find the $y$-intercept.

Q: Can a function have multiple vertical asymptotes?

A: Yes, a function can have multiple vertical asymptotes. This occurs when the denominator of the function has multiple factors that make it equal to zero.

Q: Can a function have a horizontal asymptote that is not a constant?

A: No, a function cannot have a horizontal asymptote that is not a constant. The horizontal asymptote is a constant value that the function approaches as $x$ approaches infinity or negative infinity.

Q: How do I identify a function with specific characteristics?

A: To identify a function with specific characteristics, you can use the following steps:

1. Look at the graph of the function to see if it has any vertical or horizontal asymptotes.
2. Use the steps above to find the vertical and horizontal asymptotes.
3. Use the steps above to find the $x$-intercept and $y$-intercept.
4. Compare the function to the characteristics you are looking for.

Conclusion

In conclusion, identifying functions with specific characteristics is an important skill in mathematics. By understanding the concepts of vertical and horizontal asymptotes, $x$-intercepts, and $y$-intercepts, you can identify functions with specific characteristics. We hope this article has been helpful in answering your questions about identifying functions with specific characteristics.

References

• [1] Calculus, 3rd edition, Michael Spivak
• [2] Calculus, 2nd edition, James Stewart
• [3] Mathematics, 2nd edition, Michael Artin

Future Work

In the future, we can explore other functions with specific characteristics, such as functions with multiple vertical asymptotes or functions with a horizontal asymptote that is not a constant.

Code

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the function
a = sp.symbols('a')
f = a / (x - 4)

# Find the vertical asymptote
vertical_asymptote = sp.limit(f, x, 4, '-')
print("Vertical asymptote:", vertical_asymptote)

# Find the horizontal asymptote
horizontal_asymptote = sp.limit(f, x, sp.oo)
print("Horizontal asymptote:", horizontal_asymptote)

# Find the x-intercept and y-intercept
x_intercept = f.subs(x, 0)
y_intercept = f.subs(x, 0)
print("x-intercept:", x_intercept)
print("y-intercept:", y_intercept)

This code uses the SymPy library to define the function and find the vertical asymptote, horizontal asymptote, $x$-intercept, and $y$-intercept.