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Chapter 1.1, Problem 1E

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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

Solutions

Chapter
Section
BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
To determine

Check whether the functions f(x) and g(u) are same.

Answer

The functions f and g are same.

Explanation

Given:

The functions f(x) and g(u) are defined as follows:

f(x)=x+2x (1)

g(u)=u+2u (2)

Reason:

It is known that, two functions are same if they have exactly same output values for every input values.

Obtain the domain of each function and observe.

Recall the fact that the domain is the set of all input values for which the function is defined.

Domain of f(x):

2x0x2

Thus, the domain of f(x) is (,2].

Domain of g(u):

2u0u2

Thus, the domain of g(u) is (,2].

It is clear that the functions f(x)=x+2x and g(u)=u+2u have the same domain.

For example, consider the values of x and u as 0.

In equation (1), substitute 0 for x.

f(0)=0+20=2

In equation (2), substitute 0 for u.

g(0)=0+20=2

Since the values of f(0) and g(0) are equal, the functions f(x) and g(u) are the same for 0(,2].

Therefore, it can be concluded that f(a)=g(a) for all a(,2].

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