Problem 28
Question
Find the horizontal asymptote of the graph of the function. Then sketch the graph of the function. $$ f(x)=\frac{\sqrt{x+2}}{(x+2)^{2}} $$
Step-by-Step Solution
Verified Answer
The horizontal asymptote is \( y = 0 \).
1Step 1: Identify the degrees of the function
The function is \( f(x) = \frac{\sqrt{x+2}}{(x+2)^2} \). Here, the numerator is \( \sqrt{x+2} = (x+2)^{1/2} \) and the denominator is \((x+2)^{2}\). We recognize that the highest degree of the numerator is \(1/2\) and the highest degree of the denominator is \(2\).
2Step 2: Determine the horizontal asymptote
Since the degree of the numerator is less than the degree of the denominator (\(1/2 < 2\)), the horizontal asymptote is \( y = 0 \).
3Step 3: Consider the behavior at intercepts
The graph will approach the asymptote \( y = 0 \) at both ends. Since \( f(x) eq 0 \) at any \( x \) (numerator is never zero), the function does not intercept the x-axis.
4Step 4: Analyze end-behavior of the function
As \( x \to \infty \), \( f(x) = \frac{\sqrt{x+2}}{(x+2)^2} \to 0 \). This confirms the horizontal asymptote \( y = 0 \).
5Step 5: Sketch the graph
Start by sketching the horizontal asymptote \( y = 0 \). Recognizing that the function is initially increasing as it emerges near \( x = -2 \) and eventually decreases towards zero as \( x \to \infty \), plot the curve accordingly. Ensure the graph stays above the horizontal asymptote as \( f(x) \) is positive for all \( x > -2 \).
Key Concepts
Degrees of FunctionsAsymptotic BehaviorEnd Behavior Analysis
Degrees of Functions
The degrees of functions help us understand their overall behavior, especially when concerning asymptotes and growth rates. In a rational function like \( f(x) = \frac{\sqrt{x+2}}{(x+2)^2} \), we observe different degrees for the numerator and the denominator. The numerator of this function, \( \sqrt{x+2} = (x+2)^{1/2} \), has a degree of \( 1/2 \). The denominator, \( (x+2)^2 \), has a degree of \( 2 \). Understanding these degrees is crucial, as they determine the horizontal asymptote of the function.
When the degree of the numerator is less than the degree of the denominator, as in this case, it typically results in a horizontal asymptote of \( y = 0 \). If the degrees were equal, the horizontal asymptote would be determined by the leading coefficients. But here, with \( 1/2 < 2 \), it helps us predict that the function’s long-term behavior will flatten to zero as \( x \) grows large.
When the degree of the numerator is less than the degree of the denominator, as in this case, it typically results in a horizontal asymptote of \( y = 0 \). If the degrees were equal, the horizontal asymptote would be determined by the leading coefficients. But here, with \( 1/2 < 2 \), it helps us predict that the function’s long-term behavior will flatten to zero as \( x \) grows large.
Asymptotic Behavior
Asymptotic behavior describes how a function behaves as it approaches a particular line, called an asymptote. Horizontal asymptotes tell us about the behavior of a function as \( x \) approaches infinity or negative infinity.
In the function \( f(x) = \frac{\sqrt{x+2}}{(x+2)^2} \), the horizontal asymptote is \( y = 0 \). This is because as the degree of the numerator \((1/2)\) is less than the degree of the denominator \((2)\), the values of \( f(x) \) will approach zero as \( x \) becomes very large or very small.
Even though the graph approaches \( y = 0 \), it never actually touches or crosses the horizontal asymptote in the scope of this function, as the numerator doesn't equate zero. Understanding asymptotic behavior helps in sketching the function's graph, giving an idea of where it lies as \( x \) moves toward the extreme ends.
In the function \( f(x) = \frac{\sqrt{x+2}}{(x+2)^2} \), the horizontal asymptote is \( y = 0 \). This is because as the degree of the numerator \((1/2)\) is less than the degree of the denominator \((2)\), the values of \( f(x) \) will approach zero as \( x \) becomes very large or very small.
Even though the graph approaches \( y = 0 \), it never actually touches or crosses the horizontal asymptote in the scope of this function, as the numerator doesn't equate zero. Understanding asymptotic behavior helps in sketching the function's graph, giving an idea of where it lies as \( x \) moves toward the extreme ends.
End Behavior Analysis
End behavior analysis is about finding out what happens to the value of a function as the input \( x \) approaches positive or negative infinity. This is closely related to determining horizontal asymptotes, as it tells us about the general direction the graph of a function is heading.
For \( f(x) = \frac{\sqrt{x+2}}{(x+2)^2} \), as \( x \to \infty \), the value of \( f(x) \) approaches zero, confirming the horizontal asymptote \( y = 0 \). The graph gently slopes towards this line without ever fully reaching it.
Moreover, since \( x = -2 \) is not in the domain of the function, the graph actually starts slightly to the right of \( x = -2 \), and the behavior at this point indicates the function emerges going upwards and then gradually heads downward towards zero as \( x \to \infty \). Analyzing end behavior gives us insights into how graphs behave far outside the primary turning points or critical features, essential in making a comprehensive graph sketch.
For \( f(x) = \frac{\sqrt{x+2}}{(x+2)^2} \), as \( x \to \infty \), the value of \( f(x) \) approaches zero, confirming the horizontal asymptote \( y = 0 \). The graph gently slopes towards this line without ever fully reaching it.
Moreover, since \( x = -2 \) is not in the domain of the function, the graph actually starts slightly to the right of \( x = -2 \), and the behavior at this point indicates the function emerges going upwards and then gradually heads downward towards zero as \( x \to \infty \). Analyzing end behavior gives us insights into how graphs behave far outside the primary turning points or critical features, essential in making a comprehensive graph sketch.
Other exercises in this chapter
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