# Function fusion #1

Calculus Level 3

Let $$f(x)=\int_{-1}^{x} e^{t^{2}} dt$$ and $$h(x) = f(1+g(x))$$ where $$g(x)$$ is defined for all Real $$x$$ and $$g'(x)$$ exists for all Real $$x$$. It is given that $$g(x)<0$$ for all $$x>0$$. If $$h'(1)=e$$ and $$g'(1)=1$$, then find $$g(1)$$.

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