To solve the problem of finding the function whose graph is a reflection of [tex]\(y = 3 \sqrt[3]{x}\)[/tex] about the [tex]\(x\)[/tex]-axis, let's proceed step-by-step:
1. Understanding the Original Function:
The original function given is [tex]\(y = 3 \sqrt[3]{x}\)[/tex]. Here, [tex]\(\sqrt[3]{x}\)[/tex] represents the cube root of [tex]\(x\)[/tex].
2. Reflection about the [tex]\(x\)[/tex]-Axis:
Reflecting a function about the [tex]\(x\)[/tex]-axis involves changing the sign of the [tex]\(y\)[/tex]-values. This can be done by multiplying the function by [tex]\(-1\)[/tex].
3. Applying the Reflection:
To reflect [tex]\(y = 3 \sqrt[3]{x}\)[/tex] about the [tex]\(x\)[/tex]-axis, we need to multiply the entire function by [tex]\(-1\)[/tex]:
[tex]\[
y = -1 \cdot (3 \sqrt[3]{x})
\][/tex]
4. Simplifying the Expression:
Simplify the expression obtained:
[tex]\[
y = -3 \sqrt[3]{x}
\][/tex]
Therefore, the function whose graph is the graph of [tex]\(y = 3 \sqrt[3]{x}\)[/tex] but reflected about the [tex]\(x\)[/tex]-axis is:
[tex]\[
y = -3 \sqrt[3]{x}
\][/tex]
