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Question

Find the inverse of each function. f(x) = √(2x - 3)

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Write the original function as an equation

y=2x3y=\sqrt{2x-3}

To calculate inverse, switch xx and yy then solve for y.

x=2y3x2=(2y3)2Squaring on both sidesx2=2y32y=x2+32y2=x2+32y=12x2+32\begin{align*} x&=\sqrt{2y-3}\\ \\x^2&=(\sqrt{2y-3})^2&&\text{Squaring on both sides}\\ \\x^2&=2y-3\\ \\2y&=x^2+3\\ \\\dfrac{2y}{2}&=\dfrac{x^2+3}{2}\\ \\y&=\dfrac{1}{2}x^2+\dfrac{3}{2} \end{align*}

The inverse function of f(x)=2x3f(x)=\sqrt{2x-3} is f1(x)=12x2+32f^{-1}(x)=\dfrac{1}{2}x^2+\dfrac{3}{2}

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