The first step in finding an inverse function is to write the function in “y equals” notation, as it will allow us simply to switch the variables and solve for us once again as follows:

Let y=5*tan(3x+4)

Now switch x and y:

x=5*tan(3y+4)

Now solve for y algebraically using the conventional order of
operations:

x/5=tan(3x+4)

Now take the inverse trig function, (also seen as sin^-1):

3x+4=arctan(x/5)

3x=arctan(x/5)-4

x=(arctan(x/5)-4)/3

Now, finally, we can replace y with x and vice verse:

y=(arctan(x/5)-4)/3

If you have access to a graphing utility (such as the one linked below),
the try to graph both the original and new (inverse) function to see the
relationship. Hint: it is simply a reflection around the line y=x In this case,
make sure you are graphing in radian mode.

The inverse of the function f(x) = 5*tan(3x+4) is required.

Let y = f(x) = 5*tan(3x+4)

Now express x in terms of y

=> tan (3x + 4) = y/5

=> 3x + 4 = arc tan (y/5)

=> 3x = arc tan (y/5) - 4

=> x = [arc tan (y/5) - 4]/3

interchange x and y

**y = f^-1(x) = [arc tan (x/5) - 4]/3**

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