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

Replace with y: To find the inverse function, f^(-1)(x), we first replace f(x) with y for easier manipulation. So, we have y = (2x + 3) / (2x).Swap x and y: Next, we swap x and y to find the inverse. This gives us x = (2y + 3) / (2y).Multiply by 2y: Now, we solve for y. To do this, we first multiply both sides by 2y to get rid of the fraction: 2yx = 2y + 3.Move terms: We then move all terms involving y to one side and the constant to the other side: 2yx - 2y = 3.Factor out y: Factor out y from the left side of the equation: y(2x - 2) = 3.Divide by (2x - 2): Divide both sides by (2x - 2) to solve for y: y = 3 / (2x - 2).Replace with f^(-1)(x): Replace y with f^(-1)(x) to denote that this is the inverse function: f^(-1)(x) = 3 / (2x - 2).

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Algebra 2

Simplify variable expressions using properties

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Q. For the function

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

, find

f^{-1}(x)

\newline

Answer:

f^{-1}(x)=

Replace with $y$ : To find the inverse function, $f^{-1}(x)$ , we first replace $f(x)$ with $y$ for easier manipulation. $\newline$ So, we have $y = \frac{2x + 3}{2x}$ .
Swap $x$ and $y$ : Next, we swap $x$ and $y$ to find the inverse. This gives us $x = \frac{2y + 3}{2y}$ .
Multiply by $2y$ : Now, we solve for $y$ . To do this, we first multiply both sides by $2y$ to get rid of the fraction: $\newline$ $2yx = 2y + 3$ .
Move terms: We then move all terms involving $y$ to one side and the constant to the other side: $2yx - 2y = 3$ .
Factor out $y$ : Factor out $y$ from the left side of the equation: $y(2x - 2) = 3$ .
Divide by $(2x - 2)$ : Divide both sides by $(2x - 2)$ to solve for $y$ : $y = \frac{3}{(2x - 2)}.$
Replace with $f^{-1}(x)$ : Replace $y$ with $f^{-1}(x)$ to denote that this is the inverse function: $\newline$ $f^{-1}(x) = \frac{3}{2x - 2}.$