Begin with the graph of \( y=e^{x} \) and use transformations to graph the following function. Determine the domain, range, horizontal asymptote, and \( y \)-intercept of the function. \( f(x)=-e^{x} \)

To graph the function \( f(x) = -e^x \) starting from the graph of \( y = e^x \), we can apply transformations step by step. ### Step 1: Start with the graph of \( y = e^x \) The graph of \( y = e^x \) is an exponential function that: - Passes through the point \( (0, 1) \) since \( e^0 = 1 \). - Approaches the horizontal asymptote \( y = 0 \) as \( x \to -\infty \). - Increases without bound as \( x \to \infty \). ### Step 2: Apply the transformation The function \( f(x) = -e^x \) can be obtained by reflecting the graph of \( y = e^x \) across the x-axis. This transformation changes the sign of the output values. ### Step 3: Analyze the transformed graph 1. **Domain**: The domain of \( f(x) = -e^x \) is the same as that of \( e^x \), which is all real numbers: \[ \text{Domain: } (-\infty, \infty) \] 2. **Range**: The range of \( f(x) = -e^x \) is determined by the output values. Since \( e^x \) is always positive, \( -e^x \) will always be negative. Specifically: - As \( x \to -\infty \), \( e^x \to 0 \), so \( -e^x \to 0 \) (but never reaches 0). - As \( x \to \infty \), \( e^x \to \infty \), so \( -e^x \to -\infty \). Thus, the range is: \[ \text{Range: } (-\infty, 0) \] 3. **Horizontal Asymptote**: The horizontal asymptote of \( f(x) = -e^x \) is the same as that of \( e^x \) but reflected. As \( x \to -\infty \), \( f(x) \) approaches 0 from below. Therefore, the horizontal asymptote is: \[ y = 0 \] 4. **\( y \)-intercept**: To find the \( y \)-intercept, we evaluate \( f(0) \): \[ f(0) = -e^0 = -1 \] Thus, the \( y \)-intercept is: \[ (0, -1) \] ### Summary - **Domain**: \( (-\infty, \infty) \) - **Range**: \( (-\infty, 0) \) - **Horizontal Asymptote**: \( y = 0 \) - **\( y \)-intercept**: \( (0, -1) \) The graph of \( f(x) = -e^x \) is a reflection of the graph of \( y = e^x \) across the x-axis, with the characteristics outlined above.