Find the inverse of the function f(x)=3x^(2)-27,x >= 0

Replace with y: To find the inverse of the function f(x) = 3x^2 - 27, we first replace f(x) with y to make the equation easier to work with. y = 3x^2 - 27Swap x and y: Next, we swap x and y to begin solving for the new y, which will be the inverse function. x = 3y^2 - 27Isolate y term: Now, we isolate the term containing y on one side by adding 27 to both sides of the equation. x + 27 = 3y^2Divide by 3: We then divide both sides by 3 to solve for y^2. (x + 27) / 3 = y^2Take square root: Since we are looking for y and not y^2, we take the square root of both sides. Remember that since x >= 0, we only consider the positive square root because the original function is defined for x >= 0. y = sqrt((x + 27) / 3)Rewrite inverse function: Finally, we rewrite the inverse function with the original function notation, replacing y with f^(-1)(x). f^(-1)(x) = sqrt((x + 27) / 3)

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Q. Find the inverse of the function

\newline

f(x)=3x^{2}-27,\,x \geq 0

Replace with $y$ : To find the inverse of the function $f(x) = 3x^2 - 27$ , we first replace $f(x)$ with $y$ to make the equation easier to work with. $\newline$ $y = 3x^2 - 27$
Swap x and y: Next, we swap x and y to begin solving for the new y, which will be the inverse function. $\newline$ $x = 3y^2 - 27$
Isolate y term: Now, we isolate the term containing $y$ on one side by adding $27$ to both sides of the equation. $x + 27 = 3y^2$
Divide by $3$ : We then divide both sides by $3$ to solve for $y^2$ . $(x + 27) / 3 = y^2$
Take square root: Since we are looking for $y$ and not $y^2$ , we take the square root of both sides. Remember that since $x \geq 0$ , we only consider the positive square root because the original function is defined for $x \geq 0$ . $y = \sqrt{(x + 27) / 3}$
Rewrite inverse function: Finally, we rewrite the inverse function with the original function notation, replacing $y$ with $f^{-1}(x)$ . $f^{-1}(x) = \sqrt{\frac{x + 27}{3}}$