below is the graph of $y = e^{x}$. transform it to make the graph of $y=-e^{x + 5}$. give the range and…

below is the graph of $y = e^{x}$. transform it to make the graph of $y=-e^{x + 5}$. give the range and domain of $y=-e^{x + 5}$ using interval notation.

below is the graph of $y = e^{x}$. transform it to make the graph of $y=-e^{x + 5}$. give the range and domain of $y=-e^{x + 5}$ using interval notation.

Answer

Explanation:

Step1: Identify graph - transformations

For the function $y = e^{x}$ to $y=-e^{x + 5}$, first, a horizontal shift of 5 units to the left occurs because of the $x+5$ inside the exponent. Then, a reflection about the $x -$axis occurs due to the negative sign in front of $e^{x+5}$.

Step2: Find the domain

The exponential - type function $y=-e^{x + 5}$ is defined for all real values of $x$. So, the domain of $y=-e^{x + 5}$ is all real numbers. In interval notation, the domain is $(-\infty,\infty)$.

Step3: Find the range

We know that the range of $y = e^{x}$ is $(0,\infty)$. After the reflection about the $x -$axis and the horizontal shift (the horizontal shift does not affect the range), the range of $y=-e^{x + 5}$ is $(-\infty,0)$.

Answer:

Domain: $(-\infty,\infty)$ Range: $(-\infty,0)$