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 $$ r_{y=-4} $$ $$ r_{x- ext { axis }} $$ $$ r_{y ext {-axis }} $$ Check answer Remaining Attempts : 3

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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 $$ r_{y=-4} $$ $$ r_{x-	ext { axis }} $$ $$ r_{y 	ext {-axis }} $$ Check answer Remaining Attempts : 3

UpStudy AI · Accepted Answer

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 $$ r_{x-\text{ axis}} $$.
The reflection rule used is $$ r_{x-\text{ axis}} $$.

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

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