Upper and lower bounds for $f(x)=1-e^{-tx}$ for $t>0,x\geq 0$

Question

I am trying to compute lower and upper bounds for the function $f(x)=1-e^{-tx}$ for $t>0,x\geq 0$, where the bounds should be some multiplicative of $\min(1,x)$. I tried to use the inequality $1+x\leq e^x$, but it did not lead me anywhere. Any help is much appreciated

Answer 1

It is not possible to have an inequality of the type $1-e^{-tx} \leq c \min\{{1,x}\}$ for al $t >0, x \geq 0$ with $0<c<\infty$.

If you let $t \to \infty$ this becomes $1 \leq c \min\{{1,x}\}$ which fails for $x(>0)$ close to $0$.