Explore Westonci.ca, the leading Q&A site where experts provide accurate and helpful answers to all your questions. Explore a wealth of knowledge from professionals across different disciplines on our comprehensive platform. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
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]
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]
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.