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Question 256094: please help me with this
If the function f satisfies the equation f(x+y)=f(x)+f(y) for every pair of real numbers x and y, what are the posible values of f(0)?
any real number
any positive real nuber
0 and 1 only
1 only
0 only
and why?
Answer by CharlesG2(834)   (Show Source):
You can put this solution on YOUR website! please help me with this
If the function f satisfies the equation f(x+y)=f(x)+f(y) for every pair of real numbers x and y, what are the posible values of f(0)?
any real number
any positive real nuber
0 and 1 only
1 only
0 only
and why?
Real Numbers are all numbers that can be expressed on an infinitely long number line.
Real Numbers include negative numbers ( such as -100, -33, -1 ) , zero (0) , positive numbers ( such as 1, 33, 100 ), rational numbers ( integer a / integer b where b is not 0 ) and irrational numbers ( such as pi, e, sqrt(2) ).
f(x+y)=f(x)+f(y)
example functions f(x)=x^2+x+1
f(y)=y-1
f(x)+f(y)=x^2+x+1+y+1
The two functions f(x) and g(x) can be added to make a new function h(x) where h(x)=f(x) + g(x). It is sometimes written as (f + g)(x).
here is another example:
Let f(x) = x + 3 and g(x) = 2x - 5
then (f + g)(x) = (x + 3) + (2x - 5)
so (f + g)(x) = 3x - 2
going back to original problem:
f(x+y)=f(x)+f(y)
f(0)=?
suggests x+y=0
example functions
f(x)=x^2+x+1
f(x+y)=(x+y)^2+(x+y)+1
f(0) with the above would result in 1
f(y)=y-1
f(x+y)=x+y-1
f(0) with the above would result in -1
f(x+y) = f(x) + f(y)
f(0) =f(0) + f (0)
we want f(x) + f(y) to equal 0
that means f(x) has to equal -1 * f(y) and vice versa
or f(x) and f(y) must be 0
We can only have zero only to satisfy this
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