Determine Each Feature Of The Graph Of The Given Function.$f(x)=\frac{4x+12}{x^2+x-6}$

by ADMIN 87 views

Introduction

In this article, we will explore the features of the graph of the given function f(x)=4x+12x2+x−6f(x)=\frac{4x+12}{x^2+x-6}. The graph of a function is a visual representation of the function's behavior, and it can provide valuable insights into the function's properties. We will analyze the function's domain, range, intercepts, asymptotes, and other key features to determine its graph.

Domain and Range

The domain of a function is the set of all possible input values for which the function is defined. In the case of the given function, we need to find the values of xx for which the denominator x2+x−6x^2+x-6 is not equal to zero.

import sympy as sp

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

# Define the denominator
denominator = x**2 + x - 6

# Find the values of x for which the denominator is not equal to zero
domain = sp.solve(denominator, x)

print("Domain:", domain)

The output of the above code is:

Domain: [-3, 2]

This means that the domain of the function is all real numbers except x=−3x=-3 and x=2x=2.

The range of a function is the set of all possible output values for which the function is defined. Since the function is a rational function, its range is all real numbers except the values that make the numerator equal to zero.

# Define the numerator
numerator = 4*x + 12

# Find the values of x for which the numerator is equal to zero
range_values = sp.solve(numerator, x)

print("Range values:", range_values)

The output of the above code is:

Range values: [-3]

This means that the range of the function is all real numbers except y=−12y=-12.

Intercepts

The xx-intercept of a function is the point where the function intersects the xx-axis, i.e., where y=0y=0. To find the xx-intercept of the given function, we need to set the numerator equal to zero and solve for xx.

# Define the numerator
numerator = 4*x + 12

# Find the x-intercept
x_intercept = sp.solve(numerator, x)

print("x-intercept:", x_intercept)

The output of the above code is:

x-intercept: [-3]

This means that the xx-intercept of the function is (−3,0)(-3,0).

The yy-intercept of a function is the point where the function intersects the yy-axis, i.e., where x=0x=0. To find the yy-intercept of the given function, we need to substitute x=0x=0 into the function and simplify.

# Substitute x=0 into the function
y_intercept = (4*0 + 12) / (0**2 + 0 - 6)

print("y-intercept:", y_intercept)

The output of the above code is:

y-intercept: -2

This means that the yy-intercept of the function is (0,−2)(0,-2).

Asymptotes

The vertical asymptote of a function is the vertical line that the function approaches as xx approaches a certain value. In the case of the given function, the vertical asymptote is the line x=−3x=-3 and x=2x=2.

The horizontal asymptote of a function is the horizontal line that the function approaches as xx approaches infinity. In the case of the given function, the horizontal asymptote is the line y=0y=0.

Other Key Features

The given function has a hole at x=−3x=-3 and x=2x=2 because the denominator is equal to zero at these points.

The given function has a vertical asymptote at x=−3x=-3 and x=2x=2 because the denominator is not equal to zero at these points.

The given function has a horizontal asymptote at y=0y=0 because the degree of the numerator is less than the degree of the denominator.

Conclusion

In this article, we have analyzed the features of the graph of the given function f(x)=4x+12x2+x−6f(x)=\frac{4x+12}{x^2+x-6}. We have determined the domain, range, intercepts, asymptotes, and other key features of the function. The graph of the function has a domain of all real numbers except x=−3x=-3 and x=2x=2, a range of all real numbers except y=−12y=-12, an xx-intercept of (−3,0)(-3,0), a yy-intercept of (0,−2)(0,-2), a vertical asymptote of x=−3x=-3 and x=2x=2, and a horizontal asymptote of y=0y=0. The function has a hole at x=−3x=-3 and x=2x=2 and a vertical asymptote at x=−3x=-3 and x=2x=2. The function has a horizontal asymptote at y=0y=0.

References

Code

import sympy as sp

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

# Define the denominator
denominator = x**2 + x - 6

# Find the values of x for which the denominator is not equal to zero
domain = sp.solve(denominator, x)

print("Domain:", domain)

# Define the numerator
numerator = 4*x + 12

# Find the values of x for which the numerator is equal to zero
range_values = sp.solve(numerator, x)

print("Range values:", range_values)

# Find the x-intercept
x_intercept = sp.solve(numerator, x)

print("x-intercept:", x_intercept)

# Substitute x=0 into the function
y_intercept = (4*0 + 12) / (0**2 + 0 - 6)

print("y-intercept:", y_intercept)
```<br/>
**Q&A: Determining Features of the Graph of a Function**
=====================================================

**Introduction**
---------------

In our previous article, we analyzed the features of the graph of the given function $f(x)=\frac{4x+12}{x^2+x-6}$. We determined the domain, range, intercepts, asymptotes, and other key features of the function. In this article, we will answer some frequently asked questions related to determining features of the graph of a function.

**Q: What is the domain of a function?**
--------------------------------------

A: The domain of a function is the set of all possible input values for which the function is defined. In other words, it is the set of all real numbers for which the function is valid.

**Q: How do I find the domain of a function?**
--------------------------------------------

A: To find the domain of a function, you need to find the values of $x$ for which the denominator is not equal to zero. You can use the following steps:

1. Factor the denominator.
2. Set each factor equal to zero and solve for $x$.
3. The values of $x$ that make the denominator equal to zero are not in the domain.

**Q: What is the range of a function?**
--------------------------------------

A: The range of a function is the set of all possible output values for which the function is defined. In other words, it is the set of all real numbers that the function can take as output.

**Q: How do I find the range of a function?**
--------------------------------------------

A: To find the range of a function, you need to find the values of $y$ for which the function is defined. You can use the following steps:

1. Set the numerator equal to zero and solve for $x$.
2. The values of $y$ that make the numerator equal to zero are not in the range.

**Q: What is an intercept?**
-------------------------

A: An intercept is a point where the graph of a function intersects the $x$-axis or the $y$-axis.

**Q: How do I find the intercepts of a function?**
------------------------------------------------

A: To find the intercepts of a function, you need to set the numerator equal to zero and solve for $x$ to find the $x$-intercept, and substitute $x=0$ into the function to find the $y$-intercept.

**Q: What is an asymptote?**
-------------------------

A: An asymptote is a line that the graph of a function approaches as $x$ approaches a certain value.

**Q: How do I find the asymptotes of a function?**
------------------------------------------------

A: To find the asymptotes of a function, you need to find the values of $x$ for which the denominator is equal to zero, and the values of $y$ for which the numerator is equal to zero.

**Q: What is a hole in a graph?**
---------------------------

A: A hole in a graph is a point where the graph is not defined, but the function is still valid.

**Q: How do I find the holes in a graph?**
--------------------------------------------

A: To find the holes in a graph, you need to find the values of $x$ for which the denominator is equal to zero, and the values of $y$ for which the numerator is equal to zero.

**Conclusion**
--------------

In this article, we have answered some frequently asked questions related to determining features of the graph of a function. We have discussed the domain, range, intercepts, asymptotes, and other key features of a function, and provided steps to find these features. We hope that this article has been helpful in understanding the features of a function and how to determine them.

**References**
--------------

* [1] Sympy documentation. (n.d.). Retrieved from <https://docs.sympy.org/latest/index.html>
* [2] Khan Academy. (n.d.). Retrieved from <https://www.khanacademy.org/math/algebra>

**Code**
------

```python
import sympy as sp

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

# Define the denominator
denominator = x**2 + x - 6

# Find the values of x for which the denominator is not equal to zero
domain = sp.solve(denominator, x)

print("Domain:", domain)

# Define the numerator
numerator = 4*x + 12

# Find the values of x for which the numerator is equal to zero
range_values = sp.solve(numerator, x)

print("Range values:", range_values)

# Find the x-intercept
x_intercept = sp.solve(numerator, x)

print("x-intercept:", x_intercept)

# Substitute x=0 into the function
y_intercept = (4*0 + 12) / (0**2 + 0 - 6)

print("y-intercept:", y_intercept)