For the function f(x)=(-3)/(x-2), find f^(-1)(x). Answer: f^(-1)(x)=

Rewrite function with y: To find the inverse function, f^(-1)(x), we need to switch the roles of x and y in the original function and then solve for y. Let's start by rewriting the function with y instead of f(x): y = (-3)/(x-2)Switch x and y: Now, switch x and y to find the inverse: x = (-3)/(y-2)Multiply by (y-2): Next, we solve for y. Start by multiplying both sides of the equation by (y-2) to get rid of the fraction: x(y - 2) = -3Distribute x: Distribute x on the left side of the equation: xy - 2x = -3Isolate y: Now, we want to isolate y on one side of the equation. To do this, add 2x to both sides: xy = 2x - 3Divide by x: Finally, divide both sides by x to solve for y: y = (2x - 3)/xWrite inverse function: Now that we have solved for y, we can write the inverse function: f^(-1)(x) = (2x - 3)/x

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Algebra 2

Simplify variable expressions using properties

Full solution

Q. For the function

f(x)=\frac{-3}{x-2}

, find

f^{-1}(x)

\newline

Answer:

f^{-1}(x)=

Rewrite function with y: To find the inverse function, $f^{-1}(x)$ , we need to switch the roles of $x$ and $y$ in the original function and then solve for $y$ . Let's start by rewriting the function with $y$ instead of $f(x)$ : $\newline$ $y = \frac{-3}{x-2}$
Switch x and y: Now, switch x and y to find the inverse: $x = \frac{-3}{y-2}$
Multiply by $(y-2)$ : Next, we solve for $y$ . Start by multiplying both sides of the equation by $(y-2)$ to get rid of the fraction: $\newline$ $x(y - 2) = -3$
Distribute $x$ : Distribute $x$ on the left side of the equation: $\newline$ $xy - 2x = -3$
Isolate y: Now, we want to isolate y on one side of the equation. To do this, add $2x$ to both sides: $\newline$ $xy = 2x - 3$
Divide by $x$ : Finally, divide both sides by $x$ to solve for $y$ : $y = \frac{2x - 3}{x}$
Write inverse function: Now that we have solved for $y$ , we can write the inverse function: $f^{-1}(x) = \frac{2x - 3}{x}$