let f(x) = x^3 +x^2 -3x-3 and g(x) =1/(x+1)
(i) factorize f(x) completely and find the roots of f(x) = 0
(ii) find gf and its domain
(iii) let h(x) = x^2 -3. is gf = h? justify...
(iv) find the range of gf
my answer...
(i) by long division i got f(x) = (x+1) (x^2-3)
since f(x) = 0 , the roots are -1, -sqrt(3) and sqrt(3)
(ii) gf = 1/(x+1) x (x+1) (x^2-3) = (x^2-3). thus the domain is R\{-1}
(iii) h(x) does not equal to gf because domain of h(x) is R while domain for gf is R\{-1}
(iv) i dunno. how to get range of gf??
is my answer i to iii correct?? anyone justify....
i have one question : how to become familiar to find a domain or range of a function... some function makes me confuse... from the above example, the domain is R\{-1}, according to my lecturer in order to find a range we have to refer to the domain... since the function is not defined at x = -1 than how to find the range so that i can write the range of the function is R\{something}. is my concept is correct or .... plz help me...
A function, in its simplest sense, is just a set of pairs of numbers with the condition that no two pairs have the same first member.
Of course, if you are given a formula, like f(x)= 3x+ 1, or , you can calculate the pairs: if x= 1, f(x)= 3(1)+1= 4 so a pair is (1, 4) or if x= 1 so a pair in that function is (1,1).
But, you still have to know what values you can take for x. That means that, to define a function, we would have to give a formula and the possible values for x- the domain.
If we are only given a formula, then we can look at the "maximal domain"- the set of all values of x for which the formula makes sense. But we can also define a different function having the same formula but a different domain.
For example, has maximal domain all real numbers since I can square any number. If I saw only the formula , then, by convention, I would take the maximal domain.
But I could also define with domain "all positive real numbers". That would be a different function from f(x). Or I could define with domain [0, 1]. Now h(x) is different from either f(x) or g(x).
If you were given only the formula, , without anything else said, by convention, its domain would be "all real numbers".
But, in fact, you were given which is equal to as long as x is not -1 and is not defined at x= -1. h(x) is NOT the same as f(x). They have the same "formula" but different domains.