Find the domain a) f ( x ) = 3 1 − 4 e 3...

Alissa Hancock

Alissa Hancock

Answered

2022-07-12

Find the domain
a) f ( x ) = 3 1 4 e 3 x
b) g ( x ) = ln ( 5 | x 3 | )

Answer & Explanation

Feriheashrz

Feriheashrz

Expert

2022-07-13Added 8 answers

a) Given f ( x ) = 3 1 4 e 3 x

Value of x will be excluded from the domain as the restriction of function denominator
( 1 4 e 3 x ) 0

If ( 1 4 e 3 x ) = 0
1 = 4 e 3 x 1 4 = e 3 x
log ( 1 4 ) = log ( e 3 x ) log ( 1 4 ) = 3 x log ( e )
x = log ( 1 4 ) 3 = 1 3 log ( 1 4 )

Thus, the domain of the function is ( , 1 3 log ( 1 4 ) ) ( 1 3 log ( 1 4 ) , )

b) Given g ( x ) = ln ( 5 | x 3 | )
As logarithm is only defined for positive numbers
( 5 | x 3 | ) > 0
5 > | x 3 |

And we get two conditions:
if x > 3
g ( x ) = ln ( 5 x + 3 )
g ( x ) = ln ( 8 x )

And 8 x > 0
x > 8

If x < 3
g ( x ) = ln ( 5 + x 3 )
g ( x ) = ln ( 2 + x )
And 2 + x > 0
x > 2

Thus, the domain of the function is ( 2 , 8 )

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