x = 2 e^-y + 5
2 e^-y = x-5
e^-y = (x-5)/2
ln e^-y= -y = ln [(x-5)/2]
y = - ln [(x-5)/2]
= - [ ln(x-5) - ln 2 ]
y = ln 2 - ln(x-5)
your given function is
y = 2e^-x + 5
to form the inverse, interchange the x and y variables, so the inverse is
x = 2e^-y + 5
the inverse graph will be a reflection of the original graph in the line y = x
Pick a few ordered pairs of the original function, e.g.
(0,7), (1, 5.7) , (3, 5.1) , (-1, 10.4 ) , (-5, large) , (5, just a bit over 5)
sketch the first graph
for the inverse, switch the x and y of the ordered pairs, (same as refection in line y = x)
if you want to express your inverse as a function....
x-5 = 2e^-x
take ln of both sides
ln (x-5) = ln2 + ln e^-y
ln(x-5) - ln2 = -ylne , but lne = 1
y = ln2 - ln(x-5)
y = ln(2/(x-5) )
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