Question

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

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 \( f(x) = 0 \):
\[
2\sqrt{x} = 0 \implies \sqrt{x} = 0 \implies x = 0.
\]
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 \( g(x) = 0 \):
\[
2\sqrt[3]{x} = 0 \implies \sqrt[3]{x} = 0 \implies x = 0.
\]
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 \( x \geq 0 \)), 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:
\[
x^3 - x^2 = 0 \implies x^2(x - 1) = 0.
\]
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).**