Question

Part 2. Algebraically find the inverse of the function

Answer 1

Answer:

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

Step-by-step explanation:

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

  x = f(y)

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1.

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

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2.

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

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3.

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