7. There exists a function $f ( x )$ satisfying, for all $x \in \mathbb{R}$,
A. $f ( \sin 2 x ) = \sin x$
B. $f ( \sin 2 x ) = x ^ { 2 } + x$
C. $f \left( x ^ { 2 } + 1 \right) = | x + 1 |$
D. $f \left( x ^ { 2 } + 2 x \right) = | x + 1 |$

Defined on the set of real numbers R,
$$\begin{aligned} & \beta _ { 1 } = \left\{ ( x , y ) : x ^ { 2 } + y ^ { 2 } = 1 \right\} \\ & \beta _ { 2 } = \left\{ ( x , y ) : x ^ { 2 } + y = 2 \right\} \\ & \beta _ { 3 } = \left\{ ( x , y ) : x - y ^ { 2 } = 3 \right\} \end{aligned}$$
Which of these relations define a function of the form $\mathbf { y } = \mathbf { f } ( \mathbf { x } )$ on $R$?
A) Only $\beta _ { 1 }$
B) Only $\beta _ { 2 }$
C) $\beta _ { 1 }$ and $\beta _ { 3 }$
D) $\beta _ { 2 }$ and $\beta _ { 3 }$
E) $\beta _ { 1 } , \beta _ { 2 }$ and $\beta _ { 3 }$