domain f o g ?

Question

f(x) = sqrt(2x+3)
g(x) = x^2+1
in f(x) the domain of x is [-1.5, Infinity), g(x) is all reals.

However, f o g function works out like this:
sqrt(2[x^2 + 1] + 3)
sqrt(2x^2 + 5)

A "correct answer" on an assignment has lead me to believe the domain of fog domain is all real numbers, and that the domain of f is irrelevent or superceded by fog.

Answer 1

In general, the domain of g has to be restricted to values of x which will give values of g within the domain of f. That sounds complicated, but it means simply that if g could have a value such as, say, -2, which is outside the domain of f, we'd have to cut out values of x which gave such values.

In this case, all values of g(x) are in the domain of f, so you are right, it's all real numbers.

Answer 2

Incorrect. The domain of f(x) supercedes (or restricts) the domain of f(g(x)).

It only happens that x²+1 is never less than -1.5, and that is why x can be any real number.

Answer 3

hi, that's a physically powerful question. The functionality g is a polynomial, so that's clever for all numbers, in spite of the undeniable fact that that's ok for the area of g to be constrained to a pair subset (for this reason [-3,2]). that's useful to have the flexibility to try this for purposes the place a functionality is barely valid in a undeniable selection (as an instance, in case you throw a ball, its trajectory is defined by using a parabola, yet only till it hits the floor). many times, the area of f o g incorporates all x in the area of g such that g(x) is in the area of f. that's because of the fact, to establish that (f o g)(x) = f(g(x)) to make experience, you're able to desire to have the flexibility to take g(x) (so x might desire to be in the area of g), after which you're able to desire to have the flexibility to take f of the end result (so g(x) might desire to be in the area of f). So for this reason, you're finding for all x in [-3,2] such that 2x^2 - x + a million is in [-2,5]. i'm going to assist you artwork out the respond now.

Answer 4

you are right. x can take any value in f o g.