= Solution
With the dimensionless magnetic field conjugate to the <magnetization>, $m=-\partial f_s/\partial h$. Set $y=h/|t|^\Delta$ and differentiate the homogeneous <free-energy density>:
$$
m(t,h)=|t|^{2-\alpha-\Delta}g_{m,\pm}(y),\qquad
g_{m,\pm}(y)=-g_{f,\pm}'(y).
$$
On the ordered branch, the one-sided value $g_{m,-}(0^+)$ is nonzero, so the <order-parameter critical exponent> is
$$
\boxed{\beta=2-\alpha-\Delta.}
$$
To reach the critical isotherm, take $|t|\to0$ at fixed nonzero $h$. Suppose $g_m(y)\sim A\operatorname{sgn}(y)|y|^p$ at large $|y|$. Cancellation of the residual thermal factor requires $2-\alpha-\Delta-p\Delta=0$. Hence $m(0,h)\propto\operatorname{sgn}(h)|h|^{(2-\alpha-\Delta)/\Delta}$ and
$$
\boxed{\frac1\delta=\frac{2-\alpha-\Delta}{\Delta},\qquad \Delta=\beta\delta.}
$$
The singular scaling function determines the nonanalytic critical-isotherm contribution; any analytic response background is subleading for the usual $\delta>1$.
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