Most functions do not have $[-1,1]$ as the interval on which they are to be approximated. suppose $g(t)$ is to be evaluated for $a<t<b$. Then define a new function $f(x)$ on $[-1,1]$ by $$f(x)=g\left[\frac{(b+a)+x(b-a)}{2} \right], \qquad -1\leq x\leq 1$$ Here $$t=\frac12 [(b+a)+x(b-a)]$$ represents a linear change of variable. We now approximate f(x) on [-1,1]. As a specific example, produce the cubic near-minimax approximation for $g(t)=e^t, 0\leq t\leq 1$.
My work: So my textbook only shows me how to calculate the error of the near-minimax approximation with this equation $$f(x)-c_n(x)=\frac{(x-x_0)\cdots(x-x_n)}{(n+1)!}f^{(n+1)}(c_x), \qquad -1\leq x \leq 1$$ Is this the formula to find the near-minimax approximation? I was under the impression it was for the error, but there is no other function in this sub-chapter.