Functions and Combinations of Functions Examples

Let f(x) = sin(x) and g(x) = cos(x). By combining f and g, create as many functions as possible that are continuous at every real number x.

• We can add or subtract :
  (f + g)(x) = (g + f)(x) = sin(x) + cos(x) is continuous at every real number, x.

• We can multiply:
  (fg)(x) = (gf)(x) = (sin x)(cos x) is continuous at every real number, x.

• We can divide:
  Since sin(x) is sometimes 0 and cos(x) is sometimes 0, dividing either way will result in a function that is not continuous at every real number, x. For example,  is discontinuous when , for any integer value of k.

• We can compose:
  (f ο g) = sin(cos(x)) and
  (g ο f) = cos(sin(x)) are continuous at every real number x.