Which Of The Following Expresses The Range Of Values Of $y=g(x)$, If $g(x)=\frac{5}{x+4}$?
Introduction
When dealing with rational functions, it's essential to understand the behavior of the function as the input variable changes. In this case, we're given the function $g(x)=\frac{5}{x+4}$, and we want to find the range of values that $y=g(x)$ can take. The range of a function is the set of all possible output values it can produce for the given input values.
What is a Rational Function?
A rational function is a function that can be expressed as the ratio of two polynomials. In this case, the function $g(x)=\frac{5}{x+4}$ is a rational function because it's the ratio of two polynomials: $5$ and $x+4$. Rational functions can have various types of behavior, including asymptotes, holes, and vertical tangents.
Asymptotes and Holes
To understand the behavior of the function $g(x)=\frac{5}{x+4}$, we need to examine its asymptotes and holes. An asymptote is a line that the function approaches as the input variable gets arbitrarily close to a certain value. A hole is a point where the function is not defined, but the function approaches that point as the input variable gets arbitrarily close.
Vertical Asymptote
The function $g(x)=\frac{5}{x+4}$ has a vertical asymptote at $x=-4$. This means that as $x$ approaches $-4$ from the left, the function approaches negative infinity, and as $x$ approaches $-4$ from the right, the function approaches positive infinity.
Horizontal Asymptote
The function $g(x)=\frac{5}{x+4}$ has a horizontal asymptote at $y=0$. This means that as $x$ approaches infinity, the function approaches $0$.
Range of the Function
Now that we've examined the asymptotes and holes of the function $g(x)=\frac{5}{x+4}$, we can determine the range of the function. The range of a function is the set of all possible output values it can produce for the given input values.
The Range of a Rational Function
The range of a rational function can be determined by examining the behavior of the function as the input variable changes. In this case, the function $g(x)=\frac{5}{x+4}$ has a vertical asymptote at $x=-4$ and a horizontal asymptote at $y=0$. This means that as $x$ approaches $-4$ from the left, the function approaches negative infinity, and as $x$ approaches $-4$ from the right, the function approaches positive infinity.
Determining the Range
To determine the range of the function $g(x)=\frac{5}{x+4}$, we need to examine the behavior of the function as the input variable changes. We can do this by examining the sign of the function and the behavior of the function as the input variable approaches the vertical asymptote.
Sign of the Function
The sign of the function $g(x)=\frac{5}{x+4}$ is determined by the sign of the numerator and the denominator. The numerator is always positive, and the denominator is negative when $x<-4$ and positive when $x>-4$. This means that the function is negative when $x<-4$ and positive when $x>-4$.
Behavior of the Function
As the input variable approaches the vertical asymptote at $x=-4$, the function approaches negative infinity when $x<-4$ and positive infinity when $x>-4$. This means that the function has a vertical asymptote at $x=-4$.
Range of the Function
Based on the behavior of the function as the input variable changes, we can determine the range of the function. The range of the function is the set of all possible output values it can produce for the given input values.
The Final Answer
The range of the function $g(x)=\frac{5}{x+4}$ is $(-\infty,0) \cup (0,\infty)$. This means that the function can produce any real number except $0$.
Conclusion
Introduction
In our previous article, we discussed the range of a rational function and how to determine it. In this article, we'll answer some frequently asked questions about the range of a rational function.
Q: What is the range of a rational function?
A: The range of a rational function is the set of all possible output values it can produce for the given input values.
Q: How do I determine the range of a rational function?
A: To determine the range of a rational function, you need to examine the behavior of the function as the input variable changes. This includes examining the asymptotes, holes, and sign of the function.
Q: What is a vertical asymptote?
A: A vertical asymptote is a line that the function approaches as the input variable gets arbitrarily close to a certain value. In the case of a rational function, a vertical asymptote occurs when the denominator is equal to zero.
Q: What is a horizontal asymptote?
A: A horizontal asymptote is a line that the function approaches as the input variable gets arbitrarily close to infinity. In the case of a rational function, a horizontal asymptote occurs when the degree of the numerator is less than or equal to the degree of the denominator.
Q: How do I determine the horizontal asymptote of a rational function?
A: To determine the horizontal asymptote of a rational function, you need to compare the degrees of the numerator and the denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients.
Q: What is a hole in a rational function?
A: A hole in a rational function is a point where the function is not defined, but the function approaches that point as the input variable gets arbitrarily close.
Q: How do I determine the range of a rational function with a hole?
A: To determine the range of a rational function with a hole, you need to examine the behavior of the function as the input variable approaches the hole. If the function approaches a certain value as the input variable approaches the hole, that value is not in the range of the function.
Q: Can a rational function have a range that includes zero?
A: No, a rational function cannot have a range that includes zero. This is because the denominator of a rational function cannot be zero, and if the numerator is zero, the function is not defined.
Q: Can a rational function have a range that includes negative infinity?
A: Yes, a rational function can have a range that includes negative infinity. This occurs when the function approaches negative infinity as the input variable approaches a certain value.
Q: Can a rational function have a range that includes positive infinity?
A: Yes, a rational function can have a range that includes positive infinity. This occurs when the function approaches positive infinity as the input variable approaches a certain value.
Q: How do I determine the range of a rational function with a vertical asymptote?
A: To determine the range of a rational function with a vertical asymptote, you need to examine the behavior of the function as the input variable approaches the vertical asymptote. If the function approaches a certain value as the input variable approaches the vertical asymptote, that value is not in the range of the function.
Conclusion
In conclusion, the range of a rational function is the set of all possible output values it can produce for the given input values. To determine the range of a rational function, you need to examine the behavior of the function as the input variable changes, including the asymptotes, holes, and sign of the function.