   Chapter 1.8, Problem 29E

Chapter
Section
Textbook Problem

Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain.29. h(x) = cos(1 − x2)

To determine

To find: The domain and explain the reason for the function to be continuous at every number in the domain.

Explanation

Given:

The function is h(x)=cos(1x2).

Theorems used:

Theorem 4:

If f and g are continuous at a and c is a constant, then the following functions are also continuous at a.

(i) f+g (ii) fg (iii) cf (iv) fg (v) fg if g(a)0.

Theorem 5:

(a) Any polynomial 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.

Theorem 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.

Theorem 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

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