Questions: Consider function (f). [ f(x)=-sqrt[3]x-3-1 ] Which graph represents function (f) ?

Consider function (f). [ f(x)=-sqrt[3]x-3-1 ] Which graph represents function (f) ?
Transcript text: Consider function $f$. \[ f(x)=-\sqrt[3]{x-3}-1 \] Which graph represents function $f$ ? Z. ![](https://cdn.mathpix.com/cropped/2024_08_25_8103183ebc4dd60d8cfbg.jpg?height=1112&width=1057&top_left_y=235&top_left_x=36)
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Solution

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Solution Steps

Step 1: Identify the function and its components

The given function is \( f(x) = -\sqrt[3]{x - 3} - 1 \). This is a transformation of the basic cube root function \( \sqrt[3]{x} \).

Step 2: Determine the transformations
  • The term \( x - 3 \) inside the cube root indicates a horizontal shift to the right by 3 units.
  • The negative sign in front of the cube root indicates a reflection over the x-axis.
  • The \( -1 \) outside the cube root indicates a vertical shift downward by 1 unit.
Step 3: Analyze the graph behavior
  • The cube root function \( \sqrt[3]{x} \) typically passes through the origin (0,0) and has a point of inflection there.
  • After the transformations, the point of inflection will be at (3, -1).
  • The graph should reflect over the x-axis and shift accordingly.

Final Answer

The graph that represents the function \( f(x) = -\sqrt[3]{x - 3} - 1 \) is graph W.

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