Consider a function $f:\mathbb{R}^2\rightarrow \mathbb{R}$ given by $$ \begin{equation} f(x,t)=x^3\log(t^2+1)+e^x-t. \end{equation} $$ Show that for each value of $t$ there is a function $g_t$ such that $$f(g_t(y),t)=y.$$ Also, Show that near $(x,t)=(0,0)$, the equation $f(x,t)=1$ defines $t$ as function of $x$ such that $$f(x, x(t))=1$$ near $x=0$.
MY APPROACH:
Here, $f(x,t)=0$ doesn't have any solution, also $\frac{\partial f}{\partial x}(x,t)=3x^2\log(t^2+1)+e^x\neq 0.$ Therefore, by implicit function theorem, there exist a function $g$ as function of $t$. But I am not sure, how to find $g_t(y)$ such that $$f(g_t(y),t)=y.$$
Please give me some idea, how to approach such problem if there is no solution of $f(x,t)=0$.