Questions: 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)

Solution

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.