Question

Find the related square root function, p(x)
, for the quadratic equation h(x)=4x2
with the domain of (0,∞)
.(1 point)

Answer 1

To find the related square root function \( p(x) \) for the quadratic equation \( h(x) = 4x^2 \) with the domain \( (0, \infty) \), we can follow these steps:

1. **Start with the equation:**
\[
h(x) = 4x^2
\]

2. **To express this in terms of a square root function, we can isolate \( x \):**
To find \( x \) in terms of \( h(x) \), we can take the square root of both sides. But first, let’s rewrite the equation in a usable form:
\[
h(x) = y \quad \text{(where \( y \) is the output of the function)}
\]
This gives us:
\[
y = 4x^2
\]

3. **Solving for \( x \):**
To express \( x \), we rearrange this as follows:
\[
x^2 = \frac{y}{4}
\]
Now, taking the square root of both sides:
\[
x = \sqrt{\frac{y}{4}} = \frac{\sqrt{y}}{2}
\]

4. **Relating back to \( p(x) \):**
Since \( p(x) \) will be defined in terms of \( y \), we can replace \( y \) with \( p(x) \) to give the inverse relation. Thus:
\[
p(y) = \frac{\sqrt{y}}{2}
\]

5. **Final form:**
Therefore, the related square root function \( p(y) \) to the quadratic function \( h(x) = 4x^2 \) is:
\[
p(y) = \frac{\sqrt{y}}{2}
\]

In summary, the related square root function \( p(y) \) for the quadratic equation \( h(x) = 4x^2 \) defined over the domain \( (0, \infty) \) is:
\[
p(y) = \frac{\sqrt{y}}{2}
\]