Question

Determine wheter the statement is true or false. If it istrue, explain why. If it is false, explain why or give an examplethat disproves the statement.

1) If f is a function, then f(s + t) = f(s) +f(t).

Answer 1

'False'. The statement is false because it describes a property specific to linear functions, and not all functions are linear, as illustrated by the counterexample with the function f(x) = x^2.

Step-by-step explanation:

The statement "If f is a function, then f(s + t) = f(s) + f(t)" is 'false'. This property, where f(s + t) = f(s) + f(t) for all s and t, is known as the property of additivity and it characterizes a specific type of functions called linear functions. However, not all functions have this property. For example, consider the function f(x) = x2. If we choose s = 1 and t = 2, then f(s + t) = f(3) = 9, whereas f(s) + f(t) = f(1) + f(2) = 1 + 4 = 5. Since 9 does not equal 5, this disproves the statement for the function f(x) = x2.