Why is f not a function from R to R if
a) f (x) = 1/x?
b) f (x) =√x?
c) f (x) = ±√(x^2+1)?
A function "f" from "\\R" to "\\R" is a rule that assigns to each element "x\\in \\R" exactly one
element, called "f(x), f(x)\\in \\R." in
a) "f(x)=\\dfrac{1}{x}"
"x\\not=0"
The function "f" is undefined at "x=0."
Therefore a function "f(x)=\\dfrac{1}{x}" is not a function from "\\R" to "\\R."
b) "f(x)=\\sqrt{x}"
"x\\geq0"
The function "f" is undefined for "x\\in \\R,x<0."
Therefore a function "f(x)=\\sqrt{x}" is not a function from "\\R" to "\\R."
c) "f(x)=\\pm\\sqrt{x^2+1}"
"f(0)=\\pm\\sqrt{0^2+1}=\\pm1"
A relation "f(x)=\\pm\\sqrt{x^2+1}" assigns to "x=0" two elements "-1" and "1."
Therefore "f(x)=\\pm\\sqrt{x^2+1}" is not a function from "\\R" to "\\R."
Comments
Leave a comment