State the domain and range of `y=-e^(-x)` and then find its inverse function:
The parent function is `y=e^x` which has a domain of all real numbers and a range y>0. The transformed function has been reflected across the x-axis (`y=-e^x` ) and reflected across the y-axis (`y=e^(-x)` ).
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).
The domain remains the same -- all real numbers. The range becomes y<0 since the graph has been reflected across the x-axis.
To find the inverse function:
`y=-e^(-x)` We exchange x and y and then solve for y:
`x=-e^(-y)`
`e^(-y)=-x`
`ln(e^(-y))=ln(-x)`
`-y=ln(-x)`
`y=-ln(-x)` which is the inverse function.
Note that for `y=-ln(-x)` the parent function is y=ln(x) which has a domain of x>0 and a range of all real numbers. This function has a domain of x<0 (from the reflection across the y-axis) and a range of all real numbers. (The graph of this function is a reflection of the parent function across the x-axis, but does not affect the range.)
As expected, given the domain of `f(x)=-e^(-x)` as all real numbers and the range f(x)<0, the domain of the inverse function is x<0 and the range is all real numbers.
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The domain is all real numbers and the range is y<0.
The inverse function is `y=-ln(-x)`
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The graphs of the two functions will be reflections across the line y=x:
`y=-e^(-x)` in black, y=-ln(-x) in red, and y=x in green.