Explanation Given: The function is f ( x ) = x − 3 x 1 3 . Calculation: Domain: In order to calculate the domain, take any value of x in the given function which is a real number. So clearly the domain of f is ( − ∞ , ∞ ) . Symmetry: Obtain the symmetry of the given function. Substitute the value f ( x ) = f ( − x ) . That is, f ( − x ) = ( − x ) − 3 ( − x ) 1 3 = − x + 3 ( x ) 1 3 = − ( x − 3 ( x ) 1 3 ) = − f ( x ) Hence, the given function has odd symmetry. Intercept: Let f ( x ) = 0 . That is, x − 3 x 1 3 = 0 x ( 1 − 3 x − 2 3 ) = 0 That is x = 0 . And 1 − 3 x − 2 3 = 0 3 x − 2 3 = 1 x − 2 3 = 1 3 x = ± 3 3 That is the x intercept is ( 0 , 0 ) and ( ± 3 3 , 0 ) . In order to determine the end behavior of the graph, obtain the asymptote for the given function. lim x → ± ∞ x − 3 x 1 3 = ∞ Hence, there is no asymptote. Derivatives and possible inflections: From the given function, the derivative of the given function is as follows, f ′ ( x ) = 1 − 3 ⋅ 1 3 x − 2 3 = 1 − 1 x 2 3 = 1 − 1 x 2 3 = x 2 3 − 1 x 2 3 This undefined at x = 0 and is equals to zero when at x = ± 1 . Increasing / decreasing and concavity: Assume any value of x < 0 . Take x = − 2 . That is, f ′ ( − 2 ) = ( − 2 ) 2 3 − 1 ( − 2 ) 2 3 = 1 − 1 ( − 2 ) 2 3 = 0.370 Which is greater than zero. Clearly, f ′ is positive when x is negative. Take x = − 1 2 . That is, f ′ ( − 1 2 ) = ( − 1 2 ) 2 3 − 1 ( − 1 2 ) 2 3 = 1 − 1 ( − 1 2 ) 2 3 = − 0

Single Variable Calculus: Early Transcendentals Plus MyLab Math with Pearson eText -- Access Card Package (2nd Edition) (Briggs/Cochran/Gillett Calculus 2e)

2nd Edition

ISBN: 9780321965172

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

Publisher: PEARSON

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Chapter 4.3, Problem 25E

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Single Variable Calculus: Early Transcendentals Plus MyLab Math with Pearson eText -- Access Card Package (2nd Edition) (Briggs/Cochran/Gillett Calculus 2e)

Ch. 4.1 - Prob. 1QC Ch. 4.1 - Prob. 2QC Ch. 4.1 - Sketch the graph of a function that is continuous...Ch. 4.1 - Sketch the graph of a function that has an...Ch. 4.1 - What is a critical point of a function?Ch. 4.1 - Sketch the graph of a function f that has a local...Ch. 4.1 - Sketch the graph of a function f that has a local...Ch. 4.1 - Absolute maximum/minimum values Use the following...Ch. 4.1 - Absolute maximum/minimum values Use the following...Ch. 4.1 - Absolute maximum/minimum values Use the following...

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The following function and make a graph of the function on its domain.

Solution Summary: The author illustrates the function's symmetry by subtracting the value of x in the given function.

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