We will work this exercise by first finding the inverse of the given functions. Then, we will show the graphs to verify our results.
Finding the Inverse
Recall that we can find the inverse of a function g(x) by setting y = g(x), then we switch the role of the variables and solve for y. Let's consider the first case
a, g(x) = e^(x+2)-3.
y = e^(x+2)-3 → x = e^(y+2)-3
Therefore, solving for y in x = e^(y+2)-3 will give us the inverse of g(x) = e^(x+2). It will be useful to keep in mind that the ln x and e^x are inverse functions, and hence, they undo each other.
ln e^x = x
Then, we can isolate the exponential term and use this property. Let's give it a try.
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
y =ln (x + 3)-2
We can denote the inverse as f(x).
Original Function Inverse Function
0.35cmg(x) = e^(x+2)-3 1.1cm f(x) =ln (x + 3)-2
We can follow the same procedure for the cases
b and
c since they are also . We should find the following results.
| Original Function
|
Inverse Function
|
| a. g(x) = e^(x+2)-3
|
f(x) =ln (x + 3)-2
|
| b. g(x)=- e^(x+2)+1
|
f(x) = ln (1-x)-2
|
| c. g(x)=e^(x-2)-1
|
f(x) = ln (x+1)+2
|
The rest are natural base . To find the inverse for them the procedure is similar. We set y = g(x) then, we switch the role of the variables and solve for y. Let's consider the case
d, g(x) = ln ( x+2).
y = ln( x+2) → x = ln( y+2)
Since and natural base exponential functions are inverses, we can undo a natural logarithm by using it as the argument of the
e^(ln x) = x
This will be useful to solve for y in the expression above. Let's give it a try.
x = ln(y+2)
e^x = e^(ln(y+2))
e^x = y+2
e^x -2 = y
y = e^x -2
We can denote the inverse as f(x).
Original Function Inverse Function
0.35cmg(x) = ln(x+2) 1.1cm f(x) =e^x -2
We will follow the same procedure for the cases
e and
f, since they are also natural logarithm functions. Note that we need to isolate the logarithmic term first. We should find the following results.
| Original Function
|
Inverse Function
|
| d. g(x) = ln(x+2)
|
f(x) =e^x -2
|
| e. g(x)= 2+ ln x
|
f(x) = e^(x-2)
|
| f. g(x)=ln (- x)
|
f(x) = - e^(x-2)
|
Graphing the Functions
Now we will graph the given functions together with their inverse functions to verify that we inverted them correctly. Recall that two inverse functions are the reflection of each other in the line y=x.
We can be sure that the inverse functions are correct, since for all the cases the graphs are the reflection of each other in the line y=x, as expected.
