Find largest possible domain and largest possible range of $F(x)=1+\cos2x$. Find inverse of $G(x)=x^2+2x-2, \;x \in [0,\infty)$, state it's domain.

Question

• Let $F(x)=1+\cos2x$
  Find the largest possible domain and the largest possible range of $F(x)$.

• $G(x)=x^2+2x-2, \;x \in [0, \infty)$. Find the inverse function $G^{-1}(x)$ and state it's domain.

Answer 1

Your first function is defined for any real $x$ so "largest possible domain" is all reals. Then "largest possible range' [if it means range if you use largest domain] can be found by noting the cosine part varies from $-1$ to $1$ and you're adding $1$ to that.

Second function: solve $y=x^2+2x-2$ for $x$ in terms of $y$ using quadratic equation. Whatever is under the radical needs to be zero or more, and remember you still need $x \ge 0,$ so that may further restrict $y.$ A sketch will help here.

Answer 2

Answer

thanks for help coffeemath