Evaluate the function at the given values of the independent variables. Simplify the results. g(x, y) = ln(|x + y|) (a) g(1, 0) (b) g(0, −t²) (c) g(e, 0) (d) g(e, e)

Let's evaluate the given function g ( x , y ) = ln ( ∣ x + y ∣ ) at the specified values of the independent variables.
(a) g ( 1 , 0 ) :
Substitute x = 1 and y = 0 into the function:
g ( 1 , 0 ) = ln ( ∣1 + 0∣ ) = ln ( 1 )
Since the natural logarithm of 1 is 0, we have:
g ( 1 , 0 ) = 0
(b) g ( 0 , − t 2 ) :
Substitute x = 0 and y = − t 2 into the function:
g ( 0 , − t 2 ) = ln ( ∣0 + ( − t 2 ) ∣ ) = ln ( ∣ − t 2 ∣ )
Since ∣ − t 2 ∣ = t 2 , we have:
g ( 0 , − t 2 ) = ln ( t 2 )
(c) g ( e , 0 ) :
Substitute x = e and y = 0 into the function:
g ( e , 0 ) = ln ( ∣ e + 0∣ ) = ln ( e )
Since the natural logarithm of e is 1, we have:
g ( e , 0 ) = 1
(d) g ( e , e ) :
Substitute x = e and y = e into the function:
g ( e , e ) = ln ( ∣ e + e ∣ ) = ln ( 2 e )
Using the logarithmic property ln ( ab ) = ln ( a ) + ln ( b ) , we get:
g ( e , e ) = ln ( 2 ) + ln ( e )
Since ln ( e ) = 1 , we have:
g ( e , e ) = ln ( 2 ) + 1
This evaluates the function g ( x , y ) at the given pairs of x and y . Each part is simplified based on the properties of logarithms.

Answered by MasonWilliamTurner | 2025-07-22