Let $f(x)=\sqrt{x}$. Find $g(x)$, the function that is $f(x)$ reflected over the $x$-axis and horizontally stretched by a factor of 3 .
\[
g(x)=
\]
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Answer
\(\boxed{g(x) = -\frac{\sqrt{3}}{3}\sqrt{x}}\)
Step 1 ：Let $f(x)=\sqrt{x}$
Step 2 ：Find $g(x)$, the function that is $f(x)$ reflected over the $x$-axis and horizontally stretched by a factor of 3
Step 3 ：The reflection of a function over the x-axis is achieved by taking the negative of the function
Step 4 ：The horizontal stretch by a factor of 3 is achieved by replacing x with x/3 in the function
Step 5 ：Apply these transformations to the function $f(x) = \sqrt{x}$
Step 6 ：So, $g(x) = -\sqrt{x/3}$
Step 7 ：Simplify the final answer
Step 8 ：\(\boxed{g(x) = -\frac{\sqrt{3}}{3}\sqrt{x}}\)
