jmc

A. The function $a(x)=\sqrt{2-x}$ is decreasing, so it has an inverse.

B. Note that $b(0)=b(1)=0,$ so the function $b(x)$ does not have an inverse.

C. Note that $c\left(\frac{1}{2}\right)=c(2)=\frac{5}{2},$ so the function $c(x)$ does not have an inverse.

D. The function $d(x)=2x^2+4x+7=2(x+1{)}^2+5$ is increasing on $[0,\infty ),$ so it has an inverse.

E. Note that $e(2)=e(-3)=5,$ so the function $e(x)$ does not have an inverse.

F. Both $3^x$ and $7^x$ are increasing, so $f(x)=3^x+7^x$ is also increasing. Hence, it has an inverse.

G. Suppose $g(a)=g(b)$ for some $a,$ $b>0.$ Then $$a-\frac{1}{a}=b-\frac{1}{b}.$$Multiplying both sides by $ab,$ we get $$a^{2}b-b=a{b}^2-a.$$Then $a^{2}b-a{b}^2+a-b=0,$ which factors as $(a-b)(ab+1)=0.$ Since $a$ and $b$ are positive, $ab+1$ cannot be 0, so $a=b.$

We have shown that $g(a)=g(b)$ forces $a=b,$ so the function $g(x)$ has an inverse.

H. The function $h(x)=\frac{x}{2}$ has an inverse, namely $h^{-1}(x)=2x.$

Thus, the letters of the functions that have inverses are $\boxed{\text{A, D, F, G, H}}.$