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Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

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BuyFindarrow_forward

Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516
Textbook Problem

Analyzing the Graph of a Function In Exercises 5-34, analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results.

y = x 4 x 3

To determine

To graph: The function y=x4x3

Explanation

Given: The function y=x4x3.

Graph:

The function being a rational function has a domain as all the real numbers except 3.

The function can never take a value 1 as the numerator will never be equal to the denominator. This implies that the range of the function would be (,1)(1,).

Now find the x and y intercepts by equating y and x to zero respectively to obtain:

The x-intercept is (4,0) and the y-intercept is (0,43).

Now the function does have a vertical asymptote as x=3 because at that point the denominator is not defined.

Also,

limx(x4x3)=limx(14x13x)=1limx4x1limx1x=1010=1

This implies that the function has a horizontal asymptote as y=1.

Now, differentiate the function with respect to x and equate it to zero to obtain the critical points.

1(x3)2=0

This can never be equal to zero and is not defined at 3 where the function itself is not defined. So there are no critical points.

This gives two test intervals (,3),(3,).

Let 2(,3).

f'(2)=1(23)2=125>0

The function is increasing in this interval.

Let 4(3,)

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