Homework Solution: Let f(x) = O(x) and g(x) = (x) and let c be a positive constant. Pr…

    Let f(x) = O(x) and g(x) = (x) and let c be a positive constant. Prove or disprove that f(x) + (c)g(y) = O(x+y)

    Expert Answer

     
    f(x) = O(x) means f(x) < dx, where d is some positive constant

    Permit f(x) = O(x) and g(x) = (x) and permit c be a fixed true. Prove or contradict that f(x) + (c)g(y) = O(x+y)

    Expert Acceptance

     

    f(x) = O(x) instrument f(x) < dx, where d is some fixed true

    g(x) = x, instrument g(y) = y

    Hence f(x) + (c)g(y) = f(x) + c*y
    If we re-establish f(x) < dx, then

    f(x) + (c)g(y) < dx + cy

    Here c and d are couple fixed true, If we observe that c is more than d or equals, then we can transcribe the over equation as:
    f(x) + (c)g(y) < cx + cy,
    f(x) + (c)g(y) < c(x + y), where c is equals to the true which is senior of c and d.

    Which instrument f(x) + (c)g(y)  = O(x + y), consequently c is upright a fixed true..