For the function f(x)=(3x)/(5x-4), find f^(-1)(x). Answer: f^(-1)(x)=

Define Function: 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 y = f(x), so we have: y = (3x)/(5x-4) Now switch x and y: x = (3y)/(5y-4)Switch Roles: Next, we need to solve for y. To do this, we'll multiply both sides of the equation by (5y-4) to get rid of the fraction. x(5y-4) = 3y 5xy - 4x = 3yEliminate Fraction: Now, we want to get all the terms involving y on one side of the equation and the constant term on the other side. 5xy - 3y = 4xCombine Terms: Factor out y from the left side of the equation. y(5x - 3) = 4xFactor Out: Now, divide both sides by (5x - 3) to solve for y. y = 4x / (5x - 3)Solve for y: We have found the inverse function. So, f^(-1)(x) is: f^(-1)(x) = 4x / (5x - 3)

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For the function
f(x)=(3x)/(5x-4), find
f^(-1)(x).
Answer:
f^(-1)(x)=

For the function $f(x)=\frac{3 x}{5 x-4}$ , 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}{5 x-4}

, find

f^{-1}(x)

\newline

Answer:

f^{-1}(x)=

Define Function: 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 $y = f(x)$ , so we have:
$y = \frac{3x}{5x-4}$
Now switch $x$ and $y$ :
$x = \frac{3y}{5y-4}$
Switch Roles: Next, we need to solve for $y$ . To do this, we'll multiply both sides of the equation by $(5y-4)$ to get rid of the fraction. $x(5y-4) = 3y$ $5xy - 4x = 3y$
Eliminate Fraction: Now, we want to get all the terms involving $y$ on one side of the equation and the constant term on the other side. $5xy - 3y = 4x$
Combine Terms: Factor out $y$ from the left side of the equation. $y(5x - 3) = 4x$
Factor Out: Now, divide both sides by $(5x - 3)$ to solve for $y$ . $y = \frac{4x}{5x - 3}$
Solve for y: We have found the inverse function. So, $f^{-1}(x)$ is: $\newline$ $f^{-1}(x) = \frac{4x}{5x - 3}$