Answer
The graph is shown below:
Work Step by Step
Let us consider the inequality $ y\le {{2}^{x}}$; substitute the equals symbol in place of the inequality and rewrite the equation as given below:
$ y={{2}^{x}}$
To find the value of the x-intercept, substitute y = 0 as given below:
$\begin{align}
& y={{2}^{x}} \\
& 0={{2}^{x}} \\
& x=\infty \\
\end{align}$
To find the value of the y-intercept, substitute x = 0 as given below:
$\begin{align}
& y={{2}^{x}} \\
& y={{2}^{0}} \\
& y=1 \\
\end{align}$
We know that it can be concluded from the coordinates that the inequality represents the exponential function which is continuously increasing.
Then, take origin $\left( 0,0 \right)$ as a test point and check the region in the graph to shade:
$\begin{align}
& y\le {{2}^{x}} \\
& 0\overset{?}{\mathop{\le }}\,{{2}^{0}} \\
& 0\le 1 \\
\end{align}$
We see that the above expression is correct, so the shaded region will contain the test point; that is, the region towards the origin must be shaded.
Therefore, the curve passes through point $\left( 0,1 \right)$ and never touches the x-axis (that is, the horizontal asymptote is $y=0$).
Plot the graph using the intercepts as given below:
Thus, the graph of the provided inequality is plotted and the curve passes through $\left( 0,1 \right)$. Also, we get that the shaded region contains the origin.
