Question

Part 2. Algebraically find the inverse of the function

1. f(x) = -3(x - 4)^2 - 6

2. y = sqrt (x - 4)^2 + 3

3. y = 4 cube root (x)​

Answer 1

1. f⁻¹(x) = ±²√(-⅓(x + 6)) + 4

2. y = ²√(x–3) + 4

3. y = (x/4)³

Answer 2

9514 1404 393

Answer:

1. f⁻¹(x) = 4 ±(1/3)√(-3x-18)
2. y = (x -3)² +4
3. y = x³/64

Explanation:

To find the inverse function, solve for y the equation ...

x = f(y)

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1.

`x=f(y)\\\\x=-3(y-4)^2-6\\\\x+6=-3(y-4)^2\qquad\text{add 6}\\\\\pm\sqrt{(x+6)/(-3)}=y-4\qquad\text{divide by -3, square root}\\\\4\pm(1)/(3)√(-3(x+6))=y\qquad\text{add 4, rationalize the denominator}\\\\\boxed{f^(-1)(x)=4\pm(1)/(3)√(-3x-18)}\qquad\text{simplify the radical}`

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2.

`x=√(y-4)+3\\\\x-3=√(y-4)\qquad\text{subtract 3}\\\\(x-3)^2=y-4\qquad\text{square both sides}\\\\\boxed{y=(x-3)^2+4}\qquad\text{add 4}`

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3.

`\displaystyle x=4\sqrt[3]{y}\\\\(x)/(4)=\sqrt[3]{y}\\\\\left((x)/(4)\right)^3=y\qquad\text{cube both sides}\\\\\boxed{y=(x^3)/(64)}`