Which equation is the inverse of [tex]\( y = 16x^2 + 1 \)[/tex]?

A. [tex]\( y = \pm \sqrt{\frac{x}{16} - 1} \)[/tex]
B. [tex]\( y = \frac{\pm \sqrt{x - 1}}{16} \)[/tex]
C. [tex]\( y = \frac{\pm \sqrt{x}}{4} - \frac{1}{4} \)[/tex]
D. [tex]\( y = \frac{\pm \sqrt{x - 1}}{4} \)[/tex]



Answer :

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]