For which of the following functions does g
 represent a translation of 7
 units to the right and a reflection over the x-axis of the graph of f(x)=x−−√
?



To determine which function \( g(x) \) represents a translation of 7 units to the right and a reflection over the \( x \)-axis of the graph of \( f(x) = \sqrt{x} \), let's break it down step by step:

1. **Translation by 7 units to the right**:  
   A horizontal translation by 7 units to the right is represented by replacing \( x \) with \( x - 7 \) in the function.  
   So, starting with \( f(x) = \sqrt{x} \), the function becomes:
   \[
   f(x - 7) = \sqrt{x - 7}
   \]

2. **Reflection over the \( x \)-axis**:  
   Reflecting a function over the \( x \)-axis means multiplying the entire function by \( -1 \).  
   So, reflecting \( \sqrt{x - 7} \) over the \( x \)-axis gives:
   \[
   g(x) = -\sqrt{x - 7}
   \]

Therefore, the function that represents a translation of 7 units to the right and a reflection over the \( x \)-axis is:
\[
g(x) = -\sqrt{x - 7}
\]

Let me know if you want further details or have any questions.

Here are 5 related questions for practice:
1. How would you modify \( f(x) = \sqrt{x} \) for a translation 5 units to the left and a vertical stretch by a factor of 3?
2. What is the effect of reflecting the graph of \( f(x) = x^2 \) over the y-axis?
3. How does translating \( f(x) = \sin(x) \) by 2 units up and 3 units to the right change the function?
4. What is the result of translating the function \( f(x) = |x| \) by 7 units to the left?
5. What is the new equation after reflecting \( f(x) = \ln(x) \) over the x-axis?

**Tip**: Translations along the x-axis are always done inside the function (replacing \( x \) with \( x - c \)), while translations along the y-axis are done outside the function (adding or subtracting constants directly).

For which of the following functions does g represent a translation of 7 units to the right and a reflection over the x-axis of the graph of f(x)=√x?

This problem demonstrates how to apply a translation of 7 units to the right and a reflection over the x-axis to the function f(x)=√x. Learn the step-by-step process of modifying the function and understand core concepts such as function transformations and reflections, suitable for students in Grades 9-12.

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