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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 :

GinnyAnswer
To find the inverse of the function \( y = 9x^2 - 4 \), we need to follow these steps:

1. Express \( y \) as a function of \( x \):
[tex]\[ y = 9x^2 - 4 \][/tex]

2. Replace \( y \) with \( x \) and \( x \) with \( y \) to find the inverse function:
[tex]\[ x = 9y^2 - 4 \][/tex]

3. Solve this equation for \( y \).

Start with:
[tex]\[ x = 9y^2 - 4 \][/tex]

Isolate the term with \( y^2 \):
[tex]\[ x + 4 = 9y^2 \][/tex]

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

Take the square root of both sides:
[tex]\[ y = \pm \sqrt{\frac{x + 4}{9}} \][/tex]

Since \(\sqrt{\frac{x + 4}{9}} = \frac{\sqrt{x + 4}}{\sqrt{9}}\):
[tex]\[ y = \pm \frac{\sqrt{x + 4}}{3} \][/tex]

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

Hence, the correct equation is:
[tex]\[ y = \frac{\pm \sqrt{x+4}}{3} \][/tex]

The correct choice is the third one:
[tex]\[ y = \frac{\pm \sqrt{x + 4}}{3} \][/tex]
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