To find the inverse of the given function [tex]\( y = 16x^2 + 1 \)[/tex], we follow these steps:
1. Replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] and [tex]\( x \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ x = 16y^2 + 1 \][/tex]
2. Solve for [tex]\( y \)[/tex]:
- Start by isolating the term that contains [tex]\( y \)[/tex]:
[tex]\[ x - 1 = 16y^2 \][/tex]
- Next, divide both sides by 16:
[tex]\[ \frac{x - 1}{16} = y^2 \][/tex]
- Finally, take the square root of both sides to solve for [tex]\( y \)[/tex]. Remember that taking the square root introduces both positive and negative solutions:
[tex]\[ y = \pm \sqrt{\frac{x - 1}{16}} \][/tex]
3. Simplify the expression:
- The expression under the square root can be simplified by writing:
[tex]\[ y = \pm \sqrt{\frac{x - 1}{16}} = \pm \frac{\sqrt{x - 1}}{\sqrt{16}} = \pm \frac{\sqrt{x - 1}}{4} \][/tex]
Therefore, the inverse of the given function [tex]\( y = 16x^2 + 1 \)[/tex] is:
[tex]\[ y = \pm \frac{\sqrt{x - 1}}{4} \][/tex]
This corresponds to the choice:
[tex]\[ y = \pm \sqrt{\frac{x - 1}{4}} \][/tex]
Which matches option:
[tex]\[ y = \frac{ \pm \sqrt{x-1}}{4} \][/tex]
Hence, the correct answer is:
[tex]\( \boxed{y = \frac{ \pm \sqrt{x-1}}{4}} \)[/tex]
