Question

The set of all real-valued functions f defined everywhere on the real line and such that f(1)=0 , with the operations
defined in Example 4.

Answer 1

example 4?

Answer 2

example 4 says that operation are standard addition and multiplication

Answer 3

Rephrase your question because I dont know what you want me to explain.

Answer 4

He wants to know if its a vector space or not, so lets go over the three requirements.

Answer 5

1) is f(x) = 0 in the set? well, f(x) = 0 for all x, including x = 1, so yes, it is.

Answer 6

2) if i add two functions, say f(x) and g(x), where f(1) = 0 and g(1) = 0, does f(1)+g(1) = 0? yes again, because f(1) + g(1) = 0 + 0 = 0 so this is closed under addition.

Answer 7

3) if i multiply f(x) by a scalar c, is the new function cf(x) = 0 when x = 1? we have yes again, because:
cf(1) = c*0 = 0 so its closed by scalar multiplication.

Answer 8

and because it meets these three requirements, we have ourselves a vector space! woo~

Answer 9

thanks thanks thanks sir i got the concept but sir what is this line showing what will be shape of this function