  The equivalence between $f(R)$ and scalar-tensor theories is revisited, we
consequently explored different $f(R)$ models. After consideration of specific
definition of the scalar field, we derived the potentials $V(\phi)$ for each
$f(R)$ model focusing on the early Universe, mostly the inflation epoch. For a
given potential, we applied the slow-roll approximation approach to each $f(R)$
model and obtained the expressions for the spectral index $n_{s}$ and
tensor-to-scalar ratio $r$. We determined the corresponding numerical values
associated with each of the $f(R)$ models. Our results showed that for certain
choice of parameter space, the values of $n_{s}$ and $r$ are consistent with
the Planck survey results and others produce numerical values that are in the
same range as suggested by Planck data. We further constructed the Klein-Gordon
equations $(KGE)$ of each $f(R)$ model. We found numerical solutions to each
KGE considering different values of free parameters and initial conditions of
each $f(R)$ model. All models showed that the scalar field decreases as time
increases, an indication that there is less content of the scalar field in the
late Universe.
