If f(x)=lnx and g(x)=1 / x, write the equation of f g(x) and determine the domain of f g(x).

Name: Problem 17, Chapter 51: Calculus With Trigonometry and Analytic Geometry
Uploaded: 2021-06-10T05:49:25-08:00
Duration: 4 min 8 s
Channel: Adrian Co
Description: If f(x)=lnx and g(x)=1 / x, write the equation of f g(x) and determine the domain of f g(x).

step 1 So for this problem, we're asked several things. First thing we're asked to find is the domain o

If f(x)=lnx and g(x)=1 / x, write the equation of f g(x) and...

Key Concepts

Function Composition

Function composition is the process of applying one function to the results of another. In other words, if you have two functions f and g, then the composition f(g(x)) involves first applying g to x and then applying f to the result of g(x). This process requires careful attention to the order of operations and the domains of both functions to ensure the resulting composite function is well defined.

Domain of Composite Functions

The domain of a composite function is determined by considering the domains of both the inner and outer functions. For a composite function f(g(x)), the values of x must be within the domain of g and the resulting values g(x) must lie within the domain of f. This constraint ensures that every step in the composition is valid and meaningful.

Natural Logarithm

The natural logarithm function, denoted ln(x), is defined only for positive real numbers. This restriction means that when ln(x) is used as an outer function in a composite function, its argument must be positive. Understanding this domain constraint is crucial when working with compositions involving logarithms.

Reciprocal Function

The reciprocal function, defined by 1/x, is defined for all real numbers except zero. This exclusion is pivotal when considering composite functions that include reciprocals, as any input leading to a zero denominator must be excluded from the domain to avoid undefined expressions.

Transcript

00:01 So for this problem, we're asked several things.

00:05 First thing we're asked to find is the domain of f.

00:10 So let's go ahead and look at that.

00:12 The thing about this function here is that we have the algebraic component of it and the denominator.

00:21 And when we do, just per fractions, we can't have the denominator equal to zero.

00:30 So therefore, this, when factored, is a difference of two squares.

00:35 So we have this and this that cannot equal zero.

00:41 So therefore, the domain for f for this first part is u cannot be 1 and negative 1.

00:52 Now, there are different ways of showing this.

00:56 One way to do this is with set notation.

01:00 And so you could say that this would be all you such that you cannot be negative.

01:13 One and one.

01:17 Okay, so that's one way to write that.

01:19 Another way to do it is with interval notation, so we could say that the domain is from negative infinity to negative one, union, negative one to one, union from infinity.

01:35 Just wanted to show all the different, two different ways here, just in case it's needed.

01:45 Okay, all right, so next part.

01:48 We want the domain of g of x, g equals ln of x here...

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