Q.

How do you find the inverse of arc tan(x+3) and is it a function?

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a

f-1(x)=tanx+3

b

f-1(x)=-3+tanx

c

f-1(x)=tanx-3

d

f-1(x)=-tanx-3 

answer is C.

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Detailed Solution

We know that arc tan represents the inverse trigonometric function tan-1, so let's assume the given expression as y, we have,
 y=tan-1(x+3)
As the range of  tan-1 function is (-π2, π2), so here  y  (-π2, π2).
Taking tangent function both the sides, we get,
 tan y=tan[tan-1(x+3)]
tan y=x+3                                                               [tan(tan-1x)=x]
x+3=tan y x=tan y-3                                                               ………… 1
Also, tan-1(x+3)  is a function of x, therefore we can write it as f(x), so, we get,
 f(x)= tan-1(x+3)
f(x)=y
 x=f-1(y)                                                                      ………….2
From equations (1) and (2),
 f-1(y)=tan y-3 
Replacing y with x,
 f-1(x)=tan y-3
 
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