Question

the domain of the function f(x)=sec^-1(3x-4)+tanh^-1(x+3/5)​

Answer 1

Question :

Find the domain of the function f(x)=sec^-1(3x-4)+tan h^-1(x+3/5)

Answer :

Domain of f(x) = (-8,1 ] U [5/3,2).

Given :

the function f(x)=sec^-1(3x-4)+tan h^-1(x+3/5)

To find :

The domain of the given function

Solution :

The function f(x)=sec^-1(3x-4)+tan h^-1(x+3/5)

= sec^-1(3x-4)+ 1/2 ln [{ 1 + (x+3/5)} / {1 - (x+3/5)}]

= sec^-1(3x-4)+ 1/2 ln [(8+x) / (2-x)]

For sec^-1(3x-4),

3x-4 ≤ -1 U 3x - 4 ≥ 1

= 3x ≤ 3 U 3x ≥ 5

= x ≤ U x ≥ 5/3

Hence, x ∈ (−∞, 1 ] U x ∈ [ 5/3,∞)

For ln [(8+x) / (2-x)]

(8+x) / (2-x) > 0

x ∈ (-8,0)

Hence, Domain of f(x) = (-8,1 ] U [5/3,2).

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