Question

Can I please get some help? Having a bit of trouble.

Answer 1

Well for f(x) = 2x²+2

1)
the domain is all possible values "x" can safely take, safely meaning, it has no foreseeable constraints, hmm , in this case, there aren't any, so "x" values can be ℝ, and thus the domain is ℝ

2)
the range, is all values f(x) gets, from all those mutations "x" takes, notice, you have 2x², if you make "x" some negative value like say, -1,000,000, then you'll end up with 2(-1,000,000)², which simply can be rewritten as 2(-1,000,000)(-1,000,000), now, negative times negative, is positive

that means, no matter which negative value "x" takes, the value coming out of 2x², will always be positive, and "x" is positive, the same, 2x² will spit out a positive value

so, f(x) will never be negative, since 2x²+2 will never give a negative value
the lowest f(x) can get is when x = 0, then you get 2(0)²+2, which is just 2

so, f(x) goes as low as 2, then goes back up, so the range is y ≥ 2

3)
to get the inverse of any expression, you start off by switching the variables
so, let's do so

$\bf f(x)=y=2x^2+2\qquad inverse\implies x=2y^2+2\impliedby f^{-1}(x) \\\\\\ \textit{now, let's solve for "y"} \\\\\\ x-2=2y^2\implies \cfrac{x-2}{2}=y^2\implies \sqrt{\cfrac{x-2}{2}}=y\impliedby f^{-1}(x)$

4)

the domain of the inverse, well, if two expressions are inverse of each other, say f(x) and g(x), are inverse of each other, the "domain of f(x), is the range of g(x)" and "the range of f(x) is the domain of g(x)"

so, what's the domain of the inverse? well, what's the "range" of f(x), is anything BUT below 2, well, that's also the domain for the inverse

5)

surely you know by now, what's the range of the inverse?  well, what's the domain of f(x)? is ℝ, so, that's the range of the inverse then