For the function f(x)=(2x)/(9-4x), find f^(-1)(x). Answer: f^(-1)(x)=

Replace 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 replacing f(x) with y to make it easier to work with. y = (2x) / (9 - 4x)Switch x and y: Now, switch x and y to find the inverse function. x = (2y) / (9 - 4y)Multiply both sides: Next, we need to solve for y. To do this, we'll multiply both sides of the equation by (9 - 4y) to get rid of the fraction. x * (9 - 4y) = 2yDistribute x: Distribute x on the left side of the equation. 9x - 4xy = 2yMove term -4xy: To solve for y, we need to get all the terms with y on one side of the equation. Let's move the term -4xy to the right side by adding 4xy to both sides. 9x = 2y + 4xyFactor out y: Factor out y on the right side of the equation. 9x = y(2 + 4x)Divide both sides: Now, divide both sides by (2 + 4x) to isolate y. y = 9x / (2 + 4x)Inverse function: We have found the inverse function. So, f^(-1)(x) is: f^(-1)(x) = 9x / (2 + 4x)

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Algebra 2

Simplify variable expressions using properties

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Q. For the function

f(x)=\frac{2 x}{9-4 x}

, find

f^{-1}(x)

\newline

Answer:

f^{-1}(x)=

Replace 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 replacing $f(x)$ with $y$ to make it easier to work with. $\newline$ $y = \frac{2x}{9 - 4x}$
Switch x and y: Now, switch x and y to find the inverse function. $\newline$ $x = \frac{2y}{9 - 4y}$
Multiply both sides: Next, we need to solve for $y$ . To do this, we'll multiply both sides of the equation by $(9 - 4y)$ to get rid of the fraction. $\newline$ $x \times (9 - 4y) = 2y$
Distribute $x$ : Distribute $x$ on the left side of the equation. $9x - 4xy = 2y$
Move term $-4xy$ : To solve for $y$ , we need to get all the terms with $y$ on one side of the equation. Let's move the term $-4xy$ to the right side by adding $4xy$ to both sides. $\newline$ $9x = 2y + 4xy$
Factor out y: Factor out y on the right side of the equation. $\newline$ $9x = y(2 + 4x)$
Divide both sides: Now, divide both sides by $(2 + 4x)$ to isolate $y$ . $y = \frac{9x}{2 + 4x}$
Inverse function: We have found the inverse function. So, $f^{-1}(x)$ is: $\newline$ $f^{-1}(x) = \frac{9x}{2 + 4x}$