c Homework 3_Chepter 3 (Course Work Cuestion 7, 3.1.41-Setup a Solvo LW Score: $$ 80.06 \%, 58 \mathrm{~S} $$ Pat 1 dा5 Points: 0.2 of 1 Use transformations of the graph of $$ f(x)=e^{x} $$ to graph the given function. Be sure to give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing utility to confirm the hand-drawn graphs. $$ h(x)=-e^{x} $$ Which transformation is needed to graph the function $$ h(x)=-e^{x} $$ ? Choose the correct answer below. A. The graph of $$ \bar{\Psi} x)=e^{x} $$ should be stretched vertically by a factor of -1 . B. The graph of $$ f(x)=e^{x} $$ should be reflected about the $$ y $$-axis. C. The graph of $$ f(x)=e^{x} $$ should be reflected about the $$ x $$-axis. D. The graph of $$ f(x)=e^{x} $$ shoutd be stretched horizontally by a factor of -1 .

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To determine the transformation needed to graph the function $$ h(x) = -e^{x} $$ from the base function $$ f(x) = e^{x} $$, let's analyze the options provided.

1. **Understanding the function $$ h(x) = -e^{x} $$**:
   - The function $$ f(x) = e^{x} $$ is an exponential function that increases rapidly as $$ x $$ increases and approaches 0 as $$ x $$ approaches negative infinity.
   - The negative sign in front of $$ e^{x} $$ indicates that the graph of $$ h(x) $$ will be the reflection of $$ f(x) $$ across the $$ x $$-axis.

2. **Analyzing the options**:
   - **A. The graph of $$ f(x) = e^{x} $$ should be stretched vertically by a factor of -1.**
     - This is incorrect terminology. Stretching by a factor of -1 is not a valid transformation; it implies a reflection.
   - **B. The graph of $$ f(x) = e^{x} $$ should be reflected about the $$ y $$-axis.**
     - This is incorrect. Reflecting about the $$ y $$-axis would change the function to $$ f(-x) = e^{-x} $$.
   - **C. The graph of $$ f(x) = e^{x} $$ should be reflected about the $$ x $$-axis.**
     - This is correct. Reflecting $$ f(x) = e^{x} $$ about the $$ x $$-axis gives us $$ h(x) = -e^{x} $$.
   - **D. The graph of $$ f(x) = e^{x} $$ should be stretched horizontally by a factor of -1.**
     - This is incorrect. Horizontal stretching does not apply here, and a factor of -1 would imply a reflection about the $$ y $$-axis.

3. **Conclusion**:
   - The correct transformation needed to graph $$ h(x) = -e^{x} $$ from $$ f(x) = e^{x} $$ is **C. The graph of $$ f(x) = e^{x} $$ should be reflected about the $$ x $$-axis.**

### Asymptotes, Domain, and Range

1. **Asymptotes**:
   - The function $$ h(x) = -e^{x} $$ has a horizontal asymptote at $$ y = 0 $$ (the x-axis), as $$ e^{x} $$ approaches 0 as $$ x $$ approaches negative infinity.

2. **Domain**:
   - The domain of $$ h(x) = -e^{x} $$ is all real numbers, $$ (-\infty, \infty) $$.

3. **Range**:
   - The range of $$ h(x) = -e^{x} $$ is $$ (-\infty, 0) $$ because the function takes on all negative values as $$ e^{x} $$ is always positive.

### Summary
- The transformation needed is **C**.
- Asymptote: $$ y = 0 $$.
- Domain: $$ (-\infty, \infty) $$.
- Range: $$ (-\infty, 0) $$.
The correct transformation is to reflect the graph of $$ f(x) = e^{x} $$ about the $$ x $$-axis. This results in $$ h(x) = -e^{x} $$. The function has a horizontal asymptote at $$ y = 0 $$, a domain of all real numbers, and a range of all negative real numbers.

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