Question
Find the related square root function, p(x)
, for the quadratic equation h(x)=4x2
with the domain of (0,∞)
.(1 point)
p(x)=ax−−√
where a=
But the answer is not 2
, for the quadratic equation h(x)=4x2
with the domain of (0,∞)
.(1 point)
p(x)=ax−−√
where a=
But the answer is not 2
Answers
GPT-4o mini
To find the related square root function \( p(x) \) for the quadratic function \( h(x) = 4x^2 \) with the domain \( (0, \infty) \), we can rewrite the quadratic equation in terms of \( p(x) \).
First, we isolate \( x \) in the quadratic equation:
\[
h(x) = 4x^2
\]
To express \( x \) in terms of \( h(x) \), we can solve for \( x \):
\[
x^2 = \frac{h(x)}{4}
\]
Taking the square root of both sides (and noting that we are only dealing with the domain \( (0, \infty) \)):
\[
x = \sqrt{\frac{h(x)}{4}}
\]
Now we express it in terms of \( p(x) \):
\[
p(x) = \sqrt{h(x)} = \sqrt{4x^2} = 2x
\]
Thus, we relate it to the form \( p(x) = a\sqrt{x} \).
Since \( p(x) = 2x \) implies that we are directly relating \( p(x) \) using the given quadratic structure.
If we define \( p \) as:
\[
p(x) = 2\sqrt{x}
\]
We see that:
\[
h(x) = 4x^2 = 4 \left( \frac{p(x)}{2} \right)^2 = p(x)^2
\]
Therefore, the square root function related to the quadratic \( h(x) = 4x^2 \) is:
- \( p(x) = 2\sqrt{x} \), which implies \( a = 2 \).
If your answer must differ from \( a = 2 \), then please clarify how you might be interpreting the function since typically, \( a \) would be \( 2 \) under the direct relation from \( h(x) = 4x^2 \). If the function is defined differently or interpreted differently, let me know!
First, we isolate \( x \) in the quadratic equation:
\[
h(x) = 4x^2
\]
To express \( x \) in terms of \( h(x) \), we can solve for \( x \):
\[
x^2 = \frac{h(x)}{4}
\]
Taking the square root of both sides (and noting that we are only dealing with the domain \( (0, \infty) \)):
\[
x = \sqrt{\frac{h(x)}{4}}
\]
Now we express it in terms of \( p(x) \):
\[
p(x) = \sqrt{h(x)} = \sqrt{4x^2} = 2x
\]
Thus, we relate it to the form \( p(x) = a\sqrt{x} \).
Since \( p(x) = 2x \) implies that we are directly relating \( p(x) \) using the given quadratic structure.
If we define \( p \) as:
\[
p(x) = 2\sqrt{x}
\]
We see that:
\[
h(x) = 4x^2 = 4 \left( \frac{p(x)}{2} \right)^2 = p(x)^2
\]
Therefore, the square root function related to the quadratic \( h(x) = 4x^2 \) is:
- \( p(x) = 2\sqrt{x} \), which implies \( a = 2 \).
If your answer must differ from \( a = 2 \), then please clarify how you might be interpreting the function since typically, \( a \) would be \( 2 \) under the direct relation from \( h(x) = 4x^2 \). If the function is defined differently or interpreted differently, let me know!