Problem 37
Question
Use a graphing utility to graph the function. Determine its domain and identify any vertical or horizontal asymptotes. $$g(x)=\frac{5}{x^{2}+1}$$
Step-by-Step Solution
Verified Answer
The graph of the function \(g(x)=\frac{5}{x^{2}+1}\) shows that it is a decay function that does not cross the x-axis. The domain of the function is all real numbers (\(-∞, ∞\)). The function has no vertical asymptotes. The horizontal asymptote is y=0.
1Step 1: Graphing the Function
Use a graphing calculator or online graphing tool to generate the graph of the function \(g(x)=\frac{5}{x^{2}+1}\). Observe its shape and behavior.
2Step 2: Determining the Domain
Given the function \(g(x)=\frac{5}{x^{2}+1}\), the denominator (\(x^{2}+1\)) can never equal zero since the square of a real number ( \(x^{2}\)) is always positive or zero and the added one makes the denominator always at least one. Therefore, the fraction is defined for all real numbers. Thus, the domain of the function is all real numbers, which can be written in interval notation as \(-∞, ∞\).
3Step 3: Identifying Vertical Asymptotes
The vertical asymptotes of a function are the vertical lines x = a where the function approaches positive or negative infinity as \(x\) approaches 'a'. Given the function \(g(x)=\frac{5}{x^{2}+1}\), we saw that the denominator can never be zero, therefore there are no vertical asymptotes.
4Step 4: Identifying Horizontal Asymptotes
The horizontal asymptotes of a function are the horizontal lines at y=b where the function approaches 'b' as \(x\) goes off to +∞ or -∞. Since the exponent of the denominator (\(x^{2}+1\)) of \(g(x)=\frac{5}{x^{2}+1}\) is greater than the exponent of the numerator (which is essentially 5x⁰), we know that as \(x\) goes to +∞ or -∞, the function goes to 0. So, the horizontal asymptote is y=0.
Key Concepts
Domain of a FunctionVertical AsymptotesHorizontal Asymptotes
Domain of a Function
Understanding the domain of a function is like knowing the boundaries within which a function 'plays'. For the function \(g(x)=\frac{5}{x^{2}+1}\), we determine its domain by looking for any x-values that might cause a problem, such as division by zero. In this case, since the denominator \(x^{2}+1\) will always be positive regardless of the x-value, there is no restriction, and the function 'plays' across all real numbers.
In mathematical terms, this function is defined for all real numbers because the square of any real number is non-negative. This means, in interval notation, the domain is \(-\text{∞}, \text{∞}\). Every x-value is allowed, and the function will output a real number, ensuring a graph that stretches horizontally without any interruptions or breaks.
For students grappling with domain, imagine an open field with no fences — this is your playground with endless room to run, just as the function \(g(x)\) has an endless expanse to draw its curve across the axis.
In mathematical terms, this function is defined for all real numbers because the square of any real number is non-negative. This means, in interval notation, the domain is \(-\text{∞}, \text{∞}\). Every x-value is allowed, and the function will output a real number, ensuring a graph that stretches horizontally without any interruptions or breaks.
For students grappling with domain, imagine an open field with no fences — this is your playground with endless room to run, just as the function \(g(x)\) has an endless expanse to draw its curve across the axis.
Vertical Asymptotes
Think of vertical asymptotes as invisible barriers that a function can never touch or cross. These typically occur where the function could potentially give an undefined value, like through division by zero. To pinpoint them, we look for x-values that would set the denominator of our function to zero.
For \(g(x)=\frac{5}{x^{2}+1}\), however, the denominator is \(x^{2}+1\), which is never zero since squaring a number always gives a non-negative result, and adding one keeps it positive. This means \(g(x)\) is like a wandering ghost that can travel smoothly along the xy-plane, never hitting an invisible wall because there are no vertical asymptotes to block its path.
Remember, identifying the presence of vertical asymptotes is crucial because it helps us understand where a function cannot exist and shapes the graph accordingly.
For \(g(x)=\frac{5}{x^{2}+1}\), however, the denominator is \(x^{2}+1\), which is never zero since squaring a number always gives a non-negative result, and adding one keeps it positive. This means \(g(x)\) is like a wandering ghost that can travel smoothly along the xy-plane, never hitting an invisible wall because there are no vertical asymptotes to block its path.
Remember, identifying the presence of vertical asymptotes is crucial because it helps us understand where a function cannot exist and shapes the graph accordingly.
Horizontal Asymptotes
Horizontal asymptotes are like the horizons of a function: it can get incredibly close, but it doesn't actually reach there. For rational functions, we look at the degrees of the polynomials in the numerator and denominator to find them. If the degree of the numerator is less than the degree of the denominator, as it is with \(g(x)=\frac{5}{x^{2}+1}\), the horizontal asymptote is always the x-axis, or \(y=0\).
This tells us that as \(x\) goes off to infinity in either direction, the function values get closer and closer to zero without actually getting there. It's like chasing the sunset — no matter how far you go, it's always a step ahead. This horizontal line is the asymptote and it defines the ultimate behavior of the function at extreme values of \(x\).
It's important for students to grasp that horizontal asymptotes dictate the long-term trend of a function's graph. It's the function's destiny, always approaching but forever asymptotic.
This tells us that as \(x\) goes off to infinity in either direction, the function values get closer and closer to zero without actually getting there. It's like chasing the sunset — no matter how far you go, it's always a step ahead. This horizontal line is the asymptote and it defines the ultimate behavior of the function at extreme values of \(x\).
It's important for students to grasp that horizontal asymptotes dictate the long-term trend of a function's graph. It's the function's destiny, always approaching but forever asymptotic.
Other exercises in this chapter
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