f(x)=e^x+2 -3 What is the inverse of this function?

Question

Sketch f(x) and f^-1(x) on the same grid. State the domain and the range  of both f(x) and f^-1(x). Find the equation of f^-1(x).  Given f(x)=e^x+2 -3.  The x+2 is an exponent.

Answer 1

Hi Amy,

 

f(x) = e^(x+2) - 3

 

To find the inverse of any function, follow the steps below.

 

1) rewrite the function as y= rather than f(x) =

y = e^(x+2) - 3

 

2) switch and y's and x's

x = e^(y+2) - 3

 

3) solve the equation above for y

x = e^(y+2) - 3

x+3 = e^(y+2)

ln(x+3) = ln^{e^(y+2)}

ln(x+3) = y+2

ln(x+3) - 2 = y

 

4) change y to f^-1(x)

f^-1(x) = ln(x+3) -2

 

 

Domain

f(x) : (-∞,∞)

f^-1(x): (-∞,-3) u (-3,∞)  or x≠3

 

Range 

f(x) : (-∞,∞)
f^-1(x): (-∞,∞)