Question

Let f: X - Y, where X and Y are the set of all real numbers, and x and h are real numbers. a. Find a function f such that the equation / (x + h) = / (x) + f (h) is not true for all values of x and h. Justify your reasoning. b. Find a function / such that equation / (x + h) = / (x) + / (h) is true for all values of x and h. Justify your reasoning. c. Let /(x) = 2*. Find a value for x and a value for h that makes f(x + h) = f(x) + / (h) a true number sentence.

Answer 1

a. A function f such that the equation f(x + h) = f(x) + f(h) is not true for all values of x and h is a function that is not linear. For example, a function f(x) = x^2 is not linear because it does not satisfy the equation f(x + h) = f(x) + f(h).

b. A function f such that the equation f(x + h) = f(x) + f(h) is true for all values of x and h is a linear function. For example, a function f(x) = mx + b, where m and b are constants, is a linear function because it satisfies the equation f(x + h) = f(x) + f(h).

c. Let f(x) = 2x. If we choose x = 3 and h = 2, then f(x + h) = f(3 + 2) = f(5) = 2(5) = 10, and f(x) + f(h) = f(3) + f(2) = 2(3) + 2(2) = 6 + 4 = 10. Therefore, the equation f(x + h) = f(x) + f(h) is true in this case.