Question

the function $g(x)$ is a transformation of the cube root parent function, $f(x) = \sqrt3{x}$ what function is $g(x)$?
a. $g(x) = \sqrt3{x+2}+1$
b. $g(x) = \sqrt3{x-1}+2$
c. $g(x) = \sqrt3{x+1}+2$
d. $g(x) = \sqrt3{x-2}+1$

Explanation:

Step1: Identify parent function key point

The parent function $f(x)=\sqrt[3]{x}$ has a key point at $(0,0)$ (since $\sqrt[3]{0}=0$).

Step2: Locate transformed key point

On the graph, the corresponding key point on $g(x)$ (the inflection point matching $(0,0)$ of $f(x)$) is at $(1,2)$.

Step3: Calculate horizontal shift

To move from $x=0$ to $x=1$, we shift right by 1 unit. For a function $f(x-h)$, a right shift of $h$ units means $h=1$, so this gives $\sqrt[3]{x-1}$.

Step4: Calculate vertical shift

To move from $y=0$ to $y=2$, we shift up by 2 units. Adding 2 to the function gives $\sqrt[3]{x-1}+2$.

Step5: Match to options

This matches option B.

Answer:

B. $g(x) = \sqrt[3]{x-1} + 2$