Question

Is the following function continuous on (-[infinity], [infinity])? f(x) = sin(x) + cos(x)

a) Yes
b) No
c) It depends on the value of x
d) The continuity cannot be determined

Answer 1

The function f(x) = sin(x) + cos(x) is continuous on the entire real number line, as both components are continuous and the sum of continuous functions remains continuous.

Step-by-step explanation:

The function f(x) = sin(x) + cos(x) is indeed continuous on (-∞, ∞). Both sine and cosine functions are continuous for all real numbers, and the sum of two continuous functions is also continuous.

A step by step explanation of why this function is continuous includes noting that sin(x) and cos(x) are continuous individually due to their periodic nature and having no undefined points or discontinuities. Since the sum of continuous functions retains continuity, f(x) does not depend on the value of x to be continuous; it is continuous everywhere.