Question 619389: sketch y= -log5(-x) by first describing the transformation performed on the equation y=log5(x)
find the eqution of the inverse function
sketch theinverse fuction on the same axes
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! original equation is:
y = -log(5,-x)
the value of x would have to be negative in order to make log(-x) to be positive.
the graph of this equation would be:
the inverse function of:
y = -log(5,-x) is found as follows:
multiply both sides of the equation by -1 to get:
-y = log(5,-x)
this equation is true if and only if:
5^(-y) = -x
multiply both sides of this equation by -1 to get:
-5^(-y) = x which is the same as:
x = -5^(-y)
replace x with y and y with x and you have the inverse equation of:
y = -5^(-x)
graph this equation to get:
if these 2 equations are inverse equations of each other, then they will appear to be reflections about the line y = x.
graph all 3 equations together to get:
they look like reflections about the line y = x.
we can test by taking one of the values and seeing if:
the coordinate pair (x,y) from the logarithmic equation is equal to:
the coordinate pair of (y,x) from the exponential equation.
we'll try x = -2
in the logarithmic equation, when x = -2, y = -.4306766
in the exponential equation, when x = -.4306766, y = -2
you have:
(x,y) from the logarithmic equation = (y,x) from the exponential equation.
this confirms they are inverse functions of each other.
the inverse equation undoes what the original equation does.
let f(x) = -log(5,-x)
let g(x) = -5^(-x)
if they are inverse equations, then g(f(x)) = x
we start with x = -2
we get f(x) = -.4306766
we then get g(f(x)) = g(-.4306766) which is equal to -2 which is equal to x.
since g(x) is the inverse function of f(x), then g(x) can be shown as f^(-1)(x) which means that g(x) is the inverse function of f(x).
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