calculus questions!!?

Question

a) Is f(x+y) = f(x) + f(y)? Justify your answer.

b) If f(x) = a·bx , show that f(x+c)= f(x) · bc.

c) If f(x) = mx+b show that f(x+c)= f(x)+mc

i asked my teacher and he explained to me but i still do not get it.... can somebody help me plz?!!

Answer 1

a) You prove that this is wrong by counter example:
Let f(u)=u+99.
f(x+y)=x+y+99
f(x)+(y)=x+99+y+99
These are obviously not the same.

b) f(x+c)=f(x)*bc
a*b*(x+c)=a*b*x*b*c
a*b*x+a*b*c=a*b^2*x*c
This is not equal for all a,b,c,x.
WRONG.

c) This is true:
Better write
f(u)=mu+b
to make it less confusing.
Now
f(x+c)=f(x)+mc
So u is x+c on the left side and u=x on the right side:
m(x+c)+b=mx+b+mc
mx+mc+b=mx+b+mc
This is equal.

Answer 2

a) Not necessarily

if f(x) = x^2

f(x+y) = (x+y)^2 = x^2+2xy+y^2 = f(x)+2xy+f(y)

b) f(x+c) = ab(x+c) = abx+abc = f(x)+abc (Something's wrong with the question)

c) f(x+c) = m(x+c)+b = mx+b + mc = f(x)+mc

Answer 3

a) normally NO.Exempl. f(x)= e^x f(y)=e^y f(x+y)= e^(x+y)=e^x*e^y =f(x)*f(y) and NO f(x)+f(y)
b) f(x)= a*bx .If you want f(x+c) you must put x+c in place of x

a*b(x+c)=a*bx +a*b*c = f(x) +a*b*c( Not what you wrote above)

c) f(x)= mx+b f(x+c) = m(x+c) +b = mx +b +mc = f(x)+mc
I hope you got it