For the function f(x)=(7-4x)/(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 = (7 - 4x) / (3x + 8)Switch x and y: Now, switch x and y to find the inverse: x = (7 - 4y) / (3y + 8)Eliminate denominator: Next, we need to solve for y. To do this, we'll multiply both sides of the equation by (3y + 8) to eliminate the denominator: x * (3y + 8) = 7 - 4yDistribute x: Distribute x on the left side of the equation: 3xy + 8x = 7 - 4yMove terms with y: Now, we want to get all the terms with y on one side and the constant terms on the other side. Let's move the -4y term to the left side by adding 4y to both sides: 3xy + 4y + 8x = 7Factor out y: Factor out y from the terms on the left side: y(3x + 4) + 8x = 7Isolate y: Now, isolate y by subtracting 8x from both sides: y(3x + 4) = 7 - 8xDivide both sides: Finally, divide both sides by (3x + 4) to solve for y: y = (7 - 8x) / (3x + 4)Find inverse function: We have found the inverse function. So, we can write: f^(-1)(x) = (7 - 8x) / (3x + 4)

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For the function
f(x)=(7-4x)/(3x+8), find
f^(-1)(x).
Answer:
f^(-1)(x)=

For the function $f(x)=\frac{7-4 x}{3 x+8}$ , 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{7-4 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 $y$ : $y = \frac{7 - 4x}{3x + 8}$
Switch x and y: Now, switch x and y to find the inverse: $x = \frac{7 - 4y}{3y + 8}$
Eliminate denominator: Next, we need to solve for $y$ . To do this, we'll multiply both sides of the equation by $(3y + 8)$ to eliminate the denominator: $x \cdot (3y + 8) = 7 - 4y$
Distribute $x$ : Distribute $x$ on the left side of the equation: $3xy + 8x = 7 - 4y$
Move terms with y: Now, we want to get all the terms with y on one side and the constant terms on the other side. Let's move the $-4y$ term to the left side by adding $4y$ to both sides: $\newline$ $3xy + 4y + 8x = 7$
Factor out $y$ : Factor out $y$ from the terms on the left side: $y(3x + 4) + 8x = 7$
Isolate y: Now, isolate y by subtracting $8x$ from both sides: $\newline$ $y(3x + 4) = 7 - 8x$
Divide both sides: Finally, divide both sides by $(3x + 4)$ to solve for $y$ : $y = \frac{7 - 8x}{3x + 4}$
Find inverse function: We have found the inverse function. So, we can write: $\newline$ $f^{-1}(x) = \frac{7 - 8x}{3x + 4}$