function, \( f(x)=\frac{1}{x}+4 \), is reflected. Its image is the function \( g(x)=-\frac{1}{x}-4 \). Select the rule used to reflect the fun \( r_{y=-4} \) \( r_{x-\text { axis }} \) \( r_{y \text {-axis }} \) Check answer Remaining Attempts : 3

To determine the rule used to reflect the function \( f(x) = \frac{1}{x} + 4 \) to obtain the function \( g(x) = -\frac{1}{x} - 4 \), we need to analyze the transformations involved. 1. **Identify the original function**: \[ f(x) = \frac{1}{x} + 4 \] This function is a hyperbola shifted up by 4 units. 2. **Identify the reflected function**: \[ g(x) = -\frac{1}{x} - 4 \] This function is a hyperbola that has been reflected across the x-axis (the negative sign in front of \(\frac{1}{x}\)) and then shifted down by 4 units. 3. **Determine the reflection**: - The reflection across the x-axis changes the sign of the function, which is evident in the transformation from \( f(x) \) to \( g(x) \). - The additional transformation of shifting down by 4 units is a vertical translation. 4. **Conclusion**: The primary transformation that reflects \( f(x) \) to \( g(x) \) is the reflection across the x-axis. Thus, the correct rule used to reflect the function is: \[ r_{x-\text{ axis}} \] This means the answer is \( r_{x-\text{ axis}} \).