Question

if g(x) = 2x squared - 8, find g to the -1 power(x)

Answer 1

sorry i couldnt figure out how to the squared and such lol :/

Answer 2

Assuming your question is given:\[g(x)=2x^2-8\]find \(g^{-1}(x)\).

g(x) gives you a value for g for a given value of x.
\(g^{-1}(x)\) gives you the inverse of this - i.e. given a value for g, what value of x produced this?
so you need to get an expression for x in terms of g(x). the steps are as follows:\[g(x)=2x^2-8\]\[g(x)+8=2x^2\]therefore:\[2x^2=g(x)+8\]\[x^2=\frac{g(x)+8}{2}\]\[x=\pm\sqrt{\frac{g(x)+8}{2}}\]this is the inverse function and we usually replace g(x) on the right-hand-side with x, and replace the x on the left-hand-side with \(g^{-1}(x)\) to get:\[g^{-1}(x)=\pm\sqrt{\frac{x+8}{2}}\]you can check if this is correct or not by trying some values. so lets calculate g for x=2:\[g(x)=2x^2-8\]\[g(2)=2*(2)^2-8=2*4-8=8-8=0\]now lets use this value of g to calculate \(g^{-1}\):\[g^{-1}(x)=\pm\sqrt{\frac{x+8}{2}}\]\[g^{-1}(0)=\pm\sqrt{\frac{0+8}{2}}=\pm\sqrt{\frac{8}{2}}=\pm\sqrt{4}=\pm2\]so we can see that g has the value zero for either x=2 or x=-2.