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 f(x + h) = f(x) + f(h) is not true for all values of x and h.
b. Find a function f such that equation f(x + h) = f(x) + f(h) is true for all values of x and h.
c. Let f(x) = 2x. Find a value for x and a value for h that makes f(x + h) = f(x) + f(h) a true number
sentence.
Justify your reasoning.

Answer 1

a) f(x) = x^2

b) f(x) = x

c) any pair of numbers

Explanation:

HI!

a)

an example of this kind of function is f(x) = x^2 because

f(x+h) = (x+h)^2 = x^2 + h^2 + 2 xh = f(x) + f(h) + 2xh

teherfore

f(x+h) ≠ f(x) + f(h)

other example is f(x) = x^n with n a whole number different than one

e.g.

f(x)=x^3

f(x+h) = (x+h)^3 = x^3 + h^3 + 3(x^2 h + x h^2) ≠ x^3 + h^3 = f(x) + f(h)

b)

f(x) = x is a function that actually behaves as indicated

f(x+h) = x + h = f(x) + f(h)

others examples of this kind of fucntion are given by multiplying x by any number:

f(x) = ax; f(x+h) = a(x+h) = ax + ah = f(x) + f(h)

c)

Any pair of numbers will make f(x+h) = f(x) + f(h), as mentioned in the previous section

lest consider 10 and 5

f(10+5) = 2 *(10+5) = 2*15 = 30

f(10) = 2*10 = 20

f(5) = 2*5 = 10

f(10) + f(5) = 20+10 = 30 = f(10+5)