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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain.

Theorem 4

Chapter 2.5, Problem 31E, Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its , example  1

Chapter 2.5, Problem 31E, Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its , example  2

Theorem 5

Chapter 2.5, Problem 31E, Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its , example  3

Theorem 7

Chapter 2.5, Problem 31E, Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its , example  4

Theorem 9

Chapter 2.5, Problem 31E, Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its , example  5

Chapter 2.5, Problem 31E, Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its , example  6

M ( x ) = 1 + 1 x

To determine

To state: The domain and explain that the function is continuous at every number in its domain.

Explanation

Given:

The function M(x)=1+1x.

Theorems used:

5. (a) Any polynomial function is continuous everywhere; that is, it is continuous on =(,).

(b) Any rational function is continuous wherever it is defined; that is, it is continuous on its domain.

7. The functions such as “Polynomials, rational functions, root functions, trigonometric functions, inverse trigonometric functions, exponential functions and logarithmic functions” are continuous at every number in their domains.

9. If g is continuous at a and f is continuous at g(a), then the composite function fg given by (fg)(x)=f(g(x)) is continuous at a.

Calculation:

The domain is the set of all input values of the function for which the function is real and defined.

Consider the function M(x)=1+1x.

Simplify M(x) as follows,

M(x)=x+1x

The function M(x) is defined whenever x+1x must be greater than or equal to zero.

If x+1x0, then x+10 and x>0 or x+10 and x<0.

This implies that, x>0 or x1.

That is, if x+1x0 then x(,1](0,)

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