Using the included equations, compare the intercepts and intersection points of the square root and cube root functions.

Question

f(x)=2√x and g(x)=2^3√x

Responses

They have no similar points

Both graphs share their x-and y-intercepts at (0, 0) and they also intersect at the point (1,2)

Both graphs intersect at the points (1, 2) and (0, 0), but they don't share any intercepts.

Answer 1

To compare the intercepts and intersection points of the functions $f(x)=2\sqrt{x}$ and $g(x)=2\sqrt[3]{x}$, we can analyze each function.

1. Find the intercepts:

   • For $f(x)=2\sqrt{x}$:

     • x-intercept: Set $f(x)=0$: $$2\sqrt{x}=0⟹\sqrt{x}=0⟹x=0.$$ Thus, the x-intercept is $(0,0)$.
     • y-intercept: $f(0)=2\sqrt{0}=0$. Thus, the y-intercept is also $(0,0)$.

   • For $g(x)=2\sqrt[3]{x}$:

     • x-intercept: Set $g(x)=0$: $$2\sqrt[3]{x}=0⟹\sqrt[3]{x}=0⟹x=0.$$ Thus, the x-intercept is $(0,0)$.
     • y-intercept: $g(0)=2\sqrt[3]{0}=0$. Thus, the y-intercept is also $(0,0)$.

2. Find intersection points:

   We need to set $f(x)$ equal to $g(x)$: $$2\sqrt{x}=2\sqrt[3]{x}.$$ Dividing both sides by 2 (as long as $x\ge 0$), we get: $$\sqrt{x}=\sqrt[3]{x}.$$ Raising both sides to the power of 6 (to eliminate the roots): $$x^3=x^2.$$ Rearranging gives: $$x^3-x^2=0⟹x^2(x-1)=0.$$ Thus, $x=0$ or $x=1$.

   • At $x=0$: $f(0)=g(0)=0$, so the point is $(0,0)$.
   • At $x=1$: $f(1)=2\sqrt{1}=2$ and $g(1)=2\sqrt[3]{1}=2$, so the point is $(1,2)$.

3. Conclusion:

Both functions share their x- and y-intercepts at $(0,0)$ and they also intersect at the point $(1,2)$.

Thus, the correct response is: Both graphs share their x-and y-intercepts at (0, 0) and they also intersect at the point (1, 2).