Question 1074102: Hello, I need to find g(x), the inverse function of f(x)=log_2(4x)-2. I would also like an explanation on how to approach the problem so I can try to figure out how you're supposed to solve this. Any help would be very much appreciated. Thanks!
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! let y = f(x).
you get y = log2(4x) - 2
replace y with x and x with y to get x = log2(4y) - 2
add 2 to both sides of the equation to get x + 2 = log2(4y)
by the basic definition of logarithms, x + 2 = log2(4y) if and only if 2^(x+2) = 4y
divide both sides of this equation by 4 to get (2^(x+2))/4 = y
that's the same as y = (2^(x+2)/4.
that's your inverse equation.
you know it's an inverse eqution if (x,y) from the original equaiton is equal to (y,x) from the inverse equation.
let x = 5,
f(x) = log2(4*5) - 2 which becomes f(x) = log2(20)-2
since log2(20) = log(20)/log(2), this equation becomes f(x) = log(20)/log(2) - 2
use the log function of your calculator to get log(20)/log(2) - 2 = 2.321928095
(x,y) from the original equation is equal to (5,2.321928095)
in the inverse equation, you want x to be equal to 2.321928095.
let g(x) represent the inverse equation.
your inverse equation becomes g(x) = (2^(x+2)/4.
whenb x = 2.321928095, this equation becomes g(x) = (2^(4.321928095)/4 which becomes g(x) = 5
you have (x,y) in the inverse equation is equal to (2.321928095,5)
(x,y) from the original equation is equal to (5,2.321928095)
(x,y) from the inverse equation is equal to ((2.321928095,5)
this is what they mean by (x,y) in the original equation is equal to (y,x) in the inverse equation.
the x value in the original equation becomes the y value in the inverse equation.
the y value in the original equation becomes the x value in the inverse equation.
the graph will show these equations to be reflections about the line y = x.
here's the graph.
the red line is the original equation of y = log2(4x) - 2.
the blue line is the inverse equation of y = 2^(x+2)/4
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