Which represents the reflection of $f(x)=\sqrt{x}$ over the $y$-axis?
\begin{tabular}{|c|c|}
\hline$x$ & $f(x)$ \\
\hline
\end{tabular}
Answer (1)
To reflect a function over the y-axis, replace x with − x .
Given f ( x ) = x , replace x with − x to get f ( − x ) = − x .
The reflection of the function over the y-axis is y = − x .
The final answer is y = − x .
Explanation
Understanding the Problem The problem asks us to find the reflection of the function f ( x ) = x over the y-axis. This means we need to find a new function that gives the same y-value for − x as the original function gives for x .
Finding the Reflection To reflect a function over the y-axis, we replace x with − x in the function's equation. So, if our original function is f ( x ) = x , the reflection over the y-axis will be f ( − x ) = − x .
The Reflected Function Therefore, the reflection of f ( x ) = x over the y-axis is given by the function y = − x .
Final Answer The function that represents the reflection of f ( x ) = x over the y-axis is y = − x .
Examples
Imagine you're looking at a landscape in a mirror. The mirror flips the image horizontally, creating a reflection. Similarly, in mathematics, reflecting a function over the y-axis creates a 'mirror image' of the function. For example, if you have a plant growing to the right ( f ( x ) = x ), its reflection ( f ( − x ) = − x ) would appear to be growing to the left, showing how the function's values are mirrored across the y-axis.
