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$What is the inverse of the function {:[f(x)=-(1)/(2)(x+3)?],[f^(-1)(x)=]:}$

What is the inverse of the function $f(x) = -\frac{1}{2}(x+3)$ ? $f^{-1}(x) =$

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Identify inverse functions

Full solution

Q. What is the inverse of the function

f(x) = -\frac{1}{2}(x+3)

?

f^{-1}(x) =

Switching roles and solving for y: To find the inverse of the function $f(x) = -\frac{1}{2}(x + 3)$ , we need to switch the roles of $x$ and $f(x)$ and then solve for the new $x$ . $\newline$ Let $y = f(x)$ , so we have $y = -\frac{1}{2}(x + 3)$ . $\newline$ Now we replace $f(x)$ with $y$ and solve for $x$ : $\newline$ $y = -\frac{1}{2}(x + 3)$
Switching $x$ and $y$ to find the inverse function: Next, we switch $x$ and $y$ to find the inverse function: $\newline$ $x = -\left(\frac{1}{2}\right)(y + 3)$
Solving for y to get the inverse function: Now we solve for y to get the inverse function. Start by multiplying both sides by $-2$ to get rid of the fraction: $\newline$ $-2x = y + 3$
Writing the inverse function: Next, subtract $3$ from both sides to isolate $y$ : $\newline$ $-2x - 3 = y$
Writing the inverse function: Next, subtract $3$ from both sides to isolate $y$ : $\newline$ $-2x - 3 = y$ Finally, we write the inverse function with $y$ replaced by $f^{-1}(x)$ : $\newline$ $f^{-1}(x) = -2x - 3$

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