Below is the graph of $y=e^x$.
Transform it to make the graph of $y=e^{x+6}-2$.
Give the domain and range of $y=e^{x+6}-2$ using interval notation.

Step 1 ：The transformation of the graph $y=e^x$ to $y=e^{x+6}-2$ involves two steps. First, the graph is shifted to the left by 6 units. This is represented by the $+6$ inside the exponent. Then, the graph is shifted down by 2 units. This is represented by the $-2$ outside the exponent.

Step 2 ：The domain of the function $y=e^{x+6}-2$ is all real numbers because the exponential function is defined for all real numbers.

Step 3 ：The range of the function $y=e^{x+6}-2$ is all real numbers greater than $-2$. This is because the exponential function $e^x$ is always positive, so $e^{x+6}$ is also always positive. When we subtract 2, the smallest value the function can take is $-2$.

Step 4 ：Final Answer: The domain of the function $y=e^{x+6}-2$ is all real numbers, which in interval notation is $(-\mathrm{\infty },\mathrm{\infty })$. The range of the function is all real numbers greater than $-2$, which in interval notation is $(-2,\mathrm{\infty })$. So, the domain is $\boxed{{\displaystyle (-\mathrm{\infty },\mathrm{\infty })}}$ and the range is $\boxed{{\displaystyle (-2,\mathrm{\infty })}}$.