To find the inverse of the function [tex]\( y = 16x^2 + 1 \)[/tex], we need to swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] and solve for [tex]\( y \)[/tex]. Here are the steps to find the inverse:
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 quadratic term by subtracting 1 from both sides:
[tex]\[ x - 1 = 16y^2 \][/tex]
4. Divide both sides by 16 to solve for [tex]\( y^2 \)[/tex]:
[tex]\[ \frac{x - 1}{16} = y^2 \][/tex]
5. Take the square root of both sides to solve for [tex]\( y \)[/tex]:
[tex]\[ y = \pm \sqrt{\frac{x - 1}{16}} \][/tex]
6. Simplify the square root expression:
[tex]\[ y = \pm \frac{\sqrt{x - 1}}{4} \][/tex]
Thus, the inverse equation is:
[tex]\[ y = \frac{\pm \sqrt{x-1}}{4} \][/tex]
The correct choice is:
[tex]\[ y = \frac{\pm \sqrt{x-1}}{4} \][/tex]
Therefore, among the given options, the inverse function corresponds to:
[tex]\[ y = \frac{ \pm \sqrt{x-1}}{4} \][/tex]
And that is choice 4.
