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Which equation is the inverse of [tex]\( y = 9x^2 - 4 \)[/tex]?

A. [tex]\( y = \frac{\pm \sqrt{x + 4}}{9} \)[/tex]
B. [tex]\( y = \pm \sqrt{\frac{x}{9} + 4} \)[/tex]
C. [tex]\( y = \frac{\pm \sqrt{x + 4}}{3} \)[/tex]
D. [tex]\( y = \frac{\pm \sqrt{x}}{3} + \frac{2}{3} \)[/tex]


Sagot :

To determine the inverse of the function [tex]\( y = 9x^2 - 4 \)[/tex], we need to proceed through a series of steps to express it as [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex], and then solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex].

1. Start with the original equation:
[tex]\[ y = 9x^2 - 4 \][/tex]

2. Solve for [tex]\( x \)[/tex]:
- First, isolate the term involving [tex]\( x \)[/tex]:
[tex]\[ y + 4 = 9x^2 \][/tex]

- Next, divide both sides by 9:
[tex]\[ \frac{y + 4}{9} = x^2 \][/tex]

- Then, take the square root of both sides. Remember that solving for [tex]\( x \)[/tex] involves both the positive and negative square roots:
[tex]\[ x = \pm \sqrt{\frac{y + 4}{9}} \][/tex]

3. Express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
- To write [tex]\( x \)[/tex] explicitly, we get:
[tex]\[ x = \frac{\pm \sqrt{y + 4}}{3} \][/tex]

4. Form the inverse function:
- To find the inverse function, switch [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ y = \frac{\pm \sqrt{x + 4}}{3} \][/tex]

Thus, the inverse of the function [tex]\( y = 9x^2 - 4 \)[/tex] is given by:
[tex]\[ \boxed{y = \frac{\pm \sqrt{x + 4}}{3}} \][/tex]

Checking the options provided:
- First Option: [tex]\( y = \frac{\pm \sqrt{x + 4}}{9} \)[/tex]
- Second Option: [tex]\( y = \pm \sqrt{\frac{x}{9} + 4} \)[/tex]
- Third Option: [tex]\( y = \frac{\pm \sqrt{x + 4}}{3} \)[/tex]
- Fourth Option: [tex]\( y = \frac{\pm \sqrt{x}}{3} + \frac{2}{3} \)[/tex]

Clearly, the correct answer is the third option:
[tex]\[ \boxed{y = \frac{\pm \sqrt{x + 4}}{3}} \][/tex]