To determine which equation is the inverse of [tex]\( y = 16x^2 + 1 \)[/tex], we need to follow the steps to find the inverse function.
1. Start with the given equation:
[tex]\[ y = 16x^2 + 1 \][/tex]
2. Swap [tex]\(x\)[/tex] and [tex]\(y\)[/tex] to find the inverse:
[tex]\[ x = 16y^2 + 1 \][/tex]
3. Isolate the [tex]\( y^2 \)[/tex] term:
[tex]\[ x - 1 = 16y^2 \][/tex]
4. Divide both sides by 16:
[tex]\[ \frac{x - 1}{16} = y^2 \][/tex]
5. Take the square root of both sides, remembering the [tex]\( \pm \)[/tex] due to the square root:
[tex]\[ y = \pm \sqrt{\frac{x - 1}{16}} \][/tex]
6. Simplify the expression under the square root:
[tex]\[ y = \pm \frac{\sqrt{x - 1}}{4} \][/tex]
Comparing this result with the given options, we find that the correct inverse equation is:
[tex]\[ y = \pm \frac{\sqrt{x - 1}}{4} \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{y = \pm \frac{\sqrt{x - 1}}{4}} \][/tex]
