Consider the function $f(x)=b^x$. The domain of the function is set of all real numbers. Therefore, $x \in R$.
Now $y = b ^{ x }$,
$\log y=x \log$ b. Here, $b>1$ and the input of the $\log$ cannot be 0 . Hence $y >0$.
Therefore, the range of $f(x) >0$.
Now for $x \rightarrow \infty, f ( x ) \rightarrow \infty$. Also, for $x \rightarrow-\infty, f ( x ) \rightarrow 0$. Hence for large positive real numbers, the function tends to infinity.
Similarly fo negative numbers with large magnitudes, the function tends to be 0 .
Also, $f(1)=b^1=b$ and $f(0)=b^0=1$. Therefore, $(1, b)$ and $(0,1)$ lies on the graph of the function.