Let $f'(x)>0$ and $g'(x)<0$ for all $x \in R$. Then,

(a) $f\{g(x)\}>f\{g(x+1)\}$
(b) $f\{g(x)\}>f\{g(x-1)\}$
(c) $g\{f(x)\}>g\{f(x+1)\}$
(d) $g\{f(x)\}>g\{f(x-1)\}$

(a), (c)

We have,

$f^{\prime}(x)>0$ and $g^{\prime}(x)<0$ for all $x \in R$.

$\Rightarrow f(x)$ is an increasing function and $g(x)$ is a decreasing function on $R$.

$\Rightarrow f(x-1)<f(x)<f(x+1)$ and $g(x-1)>g(x)>g(x+1)$ for all $x \in R$.

$\Rightarrow g\{f(x-1)\}>g\{f(x)\}>g\{f(x+1)\}$

and, $f\{g(x)\}>f\{g(x)\}>f\{g(x+1)\}$ for all $x \in R$

$\Rightarrow g\{f(x)\}>g\{f(x+1)\}$

and, $f\{g(x)\}>f\{g(x+1)\}$ for all $x \in R$.

Hence, options (a) and (c) are correct.