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Chapter 4: Problem 44

Sketch the graph of the function using the approach presented in this section. $$f(x)=\sqrt{\frac{x}{x+4}}$$

Short Answer

Expert verified
The graph of \(f(x)=\sqrt{\frac{x}{x+4}}\) starts from (0,0), approaches but never reaches the vertical asymptote (x=-4) on the left, and approaches but never reaches the horizontal asymptote (y=0) on the right. It also has a x and y-intercept at (0,0). The domain of the function is x greater or equal to 0, range is y from 0 to +鈭�.

Step by step solution

01

Identify the Domain of the Function

Analyze the denominator and the square root to reckon out the domain of the function. The denominator i.e., x+4, should not be zero. So, x 鈮� 鈭�4. Also for a square root function, the term under the radical should greater or equal to zero, so \(x/(x+4) >=0\). We solve these inequalities for x, and find that x must be equal or greater than 0.
02

Identify the Range of the Function

Because the square root function only returns positive values or zero, the range of f(x) will be from 0 to +鈭�.
03

Identify the y-intercept

To find y-intercept, we substitute x=0 in our function \(f(x)=\sqrt{\frac{x}{x+4}}\). So, the y-intercept is (0,0).
04

Identify the x-intercepts

To find the x-intercept, we set y or f(x) equal to zero. The solution for x will be the x-intercept. As we solve \(\sqrt{\frac{x}{x+4}} = 0\), we get x=0. So, the x-intercept is also (0,0).
05

Determine the End Behavior

As x approaches -4, the denominator approaches 0, thus the function goes to infinity. So we have vertical asymptote at x=-4. As x approaches infinity, the value of \(f(x)=\sqrt{\frac{x}{x+4}}\) is decreasing and approaches 0. So there is horizontal asymptote at y=0 and at x=-4.
06

Sketch the Graph

Now that all critical points have been discussed, use this information to sketch the graph of the function. The graph starts from (0,0), approaches but never reaches the vertical asymptote (x=-4) and approaches but never reaches the horizontal asymptote (y=0) as x tends towards positive infinity.

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