2) Consider the function \( f(x)=2 x^{2} \) and \( p(x)=\left(\frac{1}{2}\right)^{x} \). a) Restrict the domain of \( f \) so that the inverse of \( f \) will also be a function. b) Write down the equation of the inverse of \( p \) in the form \( p^{-1}(x)=\ldots \) c) If the inverse of another function is \( g^{-1}(x)=2 x-4 \). Determine the equation of \( g(x) \).
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Beyond the Answer
To ensure \( f(x) = 2x^2 \) has an inverse that is also a function, we should restrict its domain to either \( x \geq 0 \) or \( x \leq 0 \). For instance, choosing \( x \geq 0 \) makes \( f \) a one-to-one function, allowing us to find a clean inverse without ambiguity in y-values. For the function \( p(x) = \left(\frac{1}{2}\right)^{x} \), we can find the inverse by solving for \( x \) in terms of \( y \). Setting \( y = p(x) \) gives \( y = \left(\frac{1}{2}\right)^{x} \). Taking the logarithm, we find \( x = -\log_{1/2}(y) \), giving us the inverse as \( p^{-1}(x) = -\log_{1/2}(x) \).
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