Q.Find the range of the following function: $f(x)=1+2^x+4^x$.

An exponential function $f(x)=a^x$ (base $a>0$, $a\ne1$, exponent $x\in R$) has the variable in the exponent rather than the base — a fundamentally faster-growing shape than any polynomial. Its domain is always $R$ and its range is always $(0,\infty)$ (an exponential is never zero or negative). As $x\to-\infty$, $a^x\to0$, giving a horizontal asymptote at $y=0$; the function is one-one (since $a^x=a^y\implies x=y$ for $a>0,a\ne1$), though never onto $R$ (it misses every non-positive value). The special base $e\approx2.718$ (the natural exponential) has a graph of similar overall shape to any other base greater than 1. …