below is the graph of y = e^x. transform it to make the graph of y = e^(-x)+3. give the range and domain of…

Answer:

Range: $(3,+\infty)$ Domain: $(-\infty,+\infty)$

Explanation:

Step1: Analyze function transformation

The original function is $y = e^{x}$. The new - function $y=e^{-x}+3$ is obtained by reflecting $y = e^{x}$ about the y - axis (due to the negative sign in the exponent) and then shifting it up 3 units.

Step2: Find the domain

For the exponential function $y = e^{-x}+3$, the exponent $-x$ can take any real - valued input. So, the domain of $y = e^{-x}+3$ is all real numbers, which in interval notation is $(-\infty,+\infty)$.

Step3: Find the range

The exponential function $y = e^{-x}$ has a range of $(0,+\infty)$ since $e^{-x}=\frac{1}{e^{x}}$ and $e^{x}>0$ for all real $x$. When we shift $y = e^{-x}$ up 3 units to get $y = e^{-x}+3$, we add 3 to each value in the range of $y = e^{-x}$. So the range of $y = e^{-x}+3$ is $(3,+\infty)$.