Question

a cube root function is shown below.
$f(x) = -3\sqrt3{x+2}+4$
which attributes does the function exhibit? select all that apply.
☐ the graph is decreasing from left to right..
☐ domain: x=r
☐ range: y=r
☐ the graph is vertically compressed by a factor of 3.
☐ the graph is translated right 2 and up 4.
☐ the point of inflection is (-2,4).

Explanation:

Step1: Analyze monotonicity

The parent cube root function $g(x)=\sqrt[3]{x}$ is increasing. For $f(x)=-3\sqrt[3]{x+2}+4$, the negative coefficient $-3$ reflects $g(x)$ over the x-axis, so $f(x)$ is decreasing.

Step2: Find domain of $f(x)$

Cube roots are defined for all real numbers. There are no restrictions on $x+2$, so domain is $x\in\mathbb{R}$.

Step3: Find range of $f(x)$

The parent cube root has range $\mathbb{R}$. Scaling by $-3$ and shifting up 4 still results in all real numbers, so range is $y\in\mathbb{R}$.

Step4: Analyze vertical transformation

A coefficient $|a|>1$ vertically stretches, not compresses. Here $|a|=3$, so it is a vertical stretch, not compression.

Step5: Analyze horizontal translation

For $f(x)=a\sqrt[3]{x-h}+k$, $h$ is horizontal shift. Here it is $x+2=x-(-2)$, so shift left 2, not right 2.

Step6: Find inflection point

The parent inflection point is $(0,0)$. For $f(x)=-3\sqrt[3]{x-(-2)}+4$, shift left 2 and up 4, so inflection point is $(-2,4)$.

Answer:

• The graph is decreasing from left to right.
• Domain: x=R
• Range: y=R
• The point of inflection is (-2,4).