Question

f(x+h)=f(x)+f(h) true for any two numbers x and h ? If so, prove it. If not, find a function for which the statement is true and a function for which the statement is false

Answer 1

The functional property f(x+h) = f(x) + f(h) is not generally true for any function but is true for linear functions such as f(x) = ax. It is not true for other functions, like quadratic functions (e.g., f(x) = x^2) because the property fails to hold when tested.

Step-by-step explanation:

The question whether f(x+h) = f(x) + f(h) is true for any two numbers x and h is a question related to the functional property known as being additive. The property mentioned is not true for all functions, but it is a defining property for a particular class of functions known as linear functions. To show this, we can consider an example of a linear function where the property holds, such as f(x) = ax for some constant a, and an example where the property does not hold, such as a quadratic function f(x) = x2.

For the linear function f(x) = ax, we can prove that f(x+h) = f(x) + f(h):

1. f(x+h) = a(x+h) = ax + ah
2. f(x) + f(h) = ax + ah
3. Therefore, f(x+h) = ax + ah = f(x) + f(h)

For the quadratic function f(x) = x2, however, the property does not hold:

1. f(x+h) = (x+h)2 = x2 + 2xh + h2
2. f(x) + f(h) = x2 + h2
3. Since 2xh is not zero in general, f(x+h) ≠ f(x) + f(h)