Find and simplify the function values. g(x, y)=ln|x+y| (a) (2,3) (b) (5,6) (c) (e, 0) (d) (0,1)

step 1 So in this question, we've been given a function here. We've been given G of X, Y, equals the na

Find and simplify the function values. g(x, y)=ln|x+y| (a)...

Key Concepts

Domain Considerations

Due to the restriction that the logarithm function only accepts positive arguments, checking the domain becomes critical. The absolute value operation often mitigates issues with negative sums, but special attention must be given when the computed value inside the absolute value is zero, as ln(0) is undefined in the realm of real numbers.

Function Evaluation

Function evaluation involves substituting specific values into a function’s formula to compute its corresponding output. This is done by replacing the variables with provided numbers, simplifying the expression, and ensuring that all operations, such as addition and absolute value, are handled properly before applying further functions like the logarithm.

Logarithmic Function

The natural logarithm function, represented as ln, is a fundamental mathematical function that gives the power to which a constant (e) must be raised to produce a certain number. This function is only defined for positive real numbers, making its domain an important aspect when evaluating or simplifying expressions involving ln.

Absolute Value

The absolute value function returns the non-negative magnitude of a given number without regard to its sign. In contexts such as ln |x+y|, it ensures that the argument of the logarithm is non-negative, aligning with the requirement that the logarithm is only defined for positive inputs.

Transcript

00:01 So in this question, we've been given a function here.

00:04 We've been given g of x, y, equals the natural log of the absolute value of x plus y.

00:13 And we want to evaluate several values of g.

00:17 So starting with g of 2, 3.

00:21 So what this means is everywhere i see an x, i'm going to replace it with 2.

00:26 And everywhere i see a y, i replace it with three.

00:32 So i get the ln of the absolute value of 2 plus 3.

00:37 That's going to give us the natural log of.

00:41 And then i'm going to want, g of 5 comma 6.

00:46 So again, where i see x, i replace it with 5.

00:52 And where i see y, i replace it with 6.

00:56 5 plus 6 is 11.

00:59 The absolute value of 11 is 11, i'm getting the natural log of 11.

01:07 In c, we want g of e, comma, 0, right? so this is the natural log of the absolute value of e plus zero.

01:22 That's certainly just the natural log of e, but does that simplify? remember what a natural log is.

01:29 A natural log is a logarithm with a base of e, yes? and so i'm saying e raised to what power gives you e? e to the first power gives you e.

01:43 My natural log of e is 1.

01:47 Then i'm going to want g of 0 .1.

01:52 So g of 01, that gives us the natural log of the absolute value of 0 plus 1...

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