For the function f(x)=(3x)/(3x+8), 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: y = (3x)/(3x + 8)Interchange x and y: Now, interchange x and y to find the inverse: x = (3y)/(3y + 8)Multiply both sides: Next, we solve for y. To do this, we'll multiply both sides of the equation by (3y + 8) to eliminate the denominator: x * (3y + 8) = 3yDistribute x: Distribute x on the left side of the equation: 3xy + 8x = 3yMove 3xy term: To isolate y, we need to get all the terms with y on one side. Let's move the 3xy term to the right side by subtracting it from both sides: 8x = 3y - 3xyFactor out y: Factor out y on the right side of the equation: 8x = y(3 - 3x)Divide both sides: Now, divide both sides by (3 - 3x) to solve for y: y = 8x / (3 - 3x)Find inverse function: We have found the inverse function. So, the inverse function f^(-1)(x) is: f^(-1)(x) = 8x / (3 - 3x)

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For the function
f(x)=(3x)/(3x+8), find
f^(-1)(x).
Answer:
f^(-1)(x)=

For the function $f(x)=\frac{3 x}{3 x+8}$ , find $f^{-1}(x)$ . $\newline$ Answer: $f^{-1}(x)=$

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Algebra 1

Multiplication with rational exponents

Full solution

Q. For the function

f(x)=\frac{3 x}{3 x+8}

, 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 $y$ : $y = \frac{3x}{3x + 8}$
Interchange $x$ and $y$ : Now, interchange $x$ and $y$ to find the inverse: $x = \frac{3y}{3y + 8}$
Multiply both sides: Next, we solve for $y$ . To do this, we'll multiply both sides of the equation by $(3y + 8)$ to eliminate the denominator: $x \times (3y + 8) = 3y$
Distribute $x$ : Distribute $x$ on the left side of the equation: $\newline$ $3xy + 8x = 3y$
Move $3xy$ term: To isolate $y$ , we need to get all the terms with $y$ on one side. Let's move the $3xy$ term to the right side by subtracting it from both sides: $\newline$ $8x = 3y - 3xy$
Factor out $y$ : Factor out $y$ on the right side of the equation: $\newline$ $8x = y(3 - 3x)$
Divide both sides: Now, divide both sides by $(3 - 3x)$ to solve for $y$ : $y = \frac{8x}{(3 - 3x)}$
Find inverse function: We have found the inverse function. So, the inverse function $f^{-1}(x)$ is: $\newline$ $f^{-1}(x) = \frac{8x}{3 - 3x}$