Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. 27. f(x) = 29. h(t) = x + 2 33. M(x) = cos(1) 1 - e' 30. B(u)= 3u - 2 + 3u2+ 3/2u - 3 31. L(v) = v ln(1 v) = 1 + 28. g(v) X = 3v 1 v + 2 v 15 32. f(t)=e2ln(1 + 1) 34. g(t)= cos (e' - 1) 4 Theorem If f and g are continuous at a and c is a constant, then the follow- ing functions are also continuous at a: 1. f + g 2. f-9 f 4. fg g 5. if g(a) 0 3. cf 5 Theorem (a) Any polynomial is continuous everywhere; that is, it is continuous on R = (-, ). (b) Any rational function is continuous wherever it is defined; that is, it is contin- uous on its domain. 7 Theorem The following types of functions are continuous at every number in their domains: polynomials trigonometric functions exponential functions rational functions . root functions inverse trigonometric functions logarithmic functions 9 Theorem If g is continuous at a and f is continuous at g(a), then the com- posite function fog given by (fog)(x) = f(g(x)) is continuous at a.