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mzz's Question
Math
Posted 6 months ago

Let ff be the function given by f(x)=xx2−4f(x)=\frac{x}{\sqrt{x^{2}-4}}.
(a) Find the domain of ff.
(b) Write an equation for each vertical asymptote to the graph of ff.
(c) Write an equation for each horizontal asymptote to the graph of ff.
(d) Find f′(x)f^{\prime}(x).
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Answer from Sia
Posted 6 months ago
Solution by Steps
step 1
To find the domain of the function f(x)=xx2−4f(x) = \frac{x}{\sqrt{x^2 - 4}}, we need to determine where the expression under the square root is positive
step 2
The expression x2−4x^2 - 4 is positive when x2>4x^2 > 4, which implies x<−2x < -2 or x>2x > 2. Therefore, the domain of ff is (−∞,−2)∪(2,∞)(-\infty, -2) \cup (2, \infty)
step 3
To find the vertical asymptotes of f(x)=xx2−4f(x) = \frac{x}{\sqrt{x^2 - 4}}, we need to determine where the denominator is zero
step 4
The denominator x2−4\sqrt{x^2 - 4} is zero when x2−4=0x^2 - 4 = 0, which implies x=±2x = \pm 2. Therefore, the vertical asymptotes are x=−2x = -2 and x=2x = 2
step 5
To find the horizontal asymptotes of f(x)=xx2−4f(x) = \frac{x}{\sqrt{x^2 - 4}}, we need to analyze the behavior of f(x)f(x) as xx approaches ±∞\pm \infty
step 6
As x→∞x \to \infty, f(x)→xx2=x∣x∣=1f(x) \to \frac{x}{\sqrt{x^2}} = \frac{x}{|x|} = 1. As x→−∞x \to -\infty, f(x)→xx2=x∣x∣=−1f(x) \to \frac{x}{\sqrt{x^2}} = \frac{x}{|x|} = -1. Therefore, the horizontal asymptotes are y=1y = 1 and y=−1y = -1
step 7
To find the derivative of f(x)=xx2−4f(x) = \frac{x}{\sqrt{x^2 - 4}}, we use the quotient rule
step 8
Let u=xu = x and v=x2−4v = \sqrt{x^2 - 4}. Then, u′=1u' = 1 and v′=xx2−4v' = \frac{x}{\sqrt{x^2 - 4}}
step 9
Using the quotient rule, f′(x)=u′v−uv′v2=x2−4−x⋅xx2−4x2−4=x2−4−x2x2−4x2−4=−4(x2−4)3/2f'(x) = \frac{u'v - uv'}{v^2} = \frac{\sqrt{x^2 - 4} - x \cdot \frac{x}{\sqrt{x^2 - 4}}}{x^2 - 4} = \frac{\sqrt{x^2 - 4} - \frac{x^2}{\sqrt{x^2 - 4}}}{x^2 - 4} = \frac{-4}{(x^2 - 4)^{3/2}}
Answer
(a) The domain of ff is (−∞,−2)∪(2,∞)(-\infty, -2) \cup (2, \infty).
(b) The vertical asymptotes are x=−2x = -2 and x=2x = 2.
(c) The horizontal asymptotes are y=1y = 1 and y=−1y = -1.
(d) The derivative of ff is f′(x)=−4(x2−4)3/2f'(x) = \frac{-4}{(x^2 - 4)^{3/2}}.
Key Concept
Domain and Asymptotes of Rational Functions
Explanation
The domain of a function involving a square root in the denominator is determined by where the expression under the square root is positive. Vertical asymptotes occur where the denominator is zero, and horizontal asymptotes are found by analyzing the behavior of the function as xx approaches infinity.

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