For each pair of functions in Exercises 17-34 : A. Find the domain of f, g, f+g, f-g, f g, f f,

For each pair of functions in Exercises 17-34 : A. Find the...

Key Concepts

Composition of Functions

When composing functions, indicated by notations like (f?f)(x) or f(f(x)), the domain of the composite function consists of all values x that first belong to the domain of the inner function and then, after substitution, yield values that are within the domain of the outer function. This sequential filtering process is crucial when working with compositions, especially if the functions have domain restrictions.

Arithmetic Operations on Functions

Defining the sum, difference, and product of two functions is done pointwise; that is, (f+g)(x)=f(x)+g(x), (f?g)(x)=f(x)?g(x), and (f?g)(x)=f(x)?g(x). Each resulting function inherits its domain from the requirement that both f(x) and g(x) must be defined, hence the importance of the individual function domains.

Division of Functions and Domain Considerations

When one function is divided by another to form a new function, such as (f/g)(x)=f(x)/g(x), the domain is obtained not only by the intersection of the domains of f and g but also by excluding any values where the denominator function g(x) equals zero. This additional caveat ensures that the division is defined, adhering to the principles of real number arithmetic.

Intersection of Domains in Function Operations

When performing operations such as addition, subtraction, and multiplication on two functions, the resulting function will typically be defined only where both original functions are defined. This means finding the domain of such composite operations involves taking the intersection of the individual domains of the functions involved.

Domain of a Function

The domain of a function comprises all the input values for which the function is defined. When determining a domain, one must consider any restrictions due to radicals, denominators, logarithms, or other operations that might be undefined for certain values. This forms the basis for many problems in function analysis.

Square Root Function Restrictions

Functions involving square roots, like f(x)=?(x?3) or g(x)=?(x+3), have radicands that must be non?negative because the square root of a negative number is not defined in the real number system. Consequently, solving inequalities that set the expression under the square root greater than or equal to zero is essential in determining their domains.

Transcript

00:01 Okay, so here we have that f of x is equal to the square root of x minus three, and g of x is equal to the square root of x plus three.

00:08 So the domain for f, well, we have to have that what's under the radican here has to be greater than equal to zero.

00:16 So therefore, the domain of f is going to be three to infinity, including three, right? and then the domain of g is going to be negative three.

00:29 So here's for g, we have negative 3, again, including negative 3 to infinity.

00:36 All right, so there's the domain of f and the domain of g.

00:39 Then we have the domain of f plus g is just going to be the intersection as well as f minus g and as well as f times g.

00:51 So for f plus g and f plus g is going to be again, three to infinity.

00:57 So here is f plus g, f minus g, and f times g.

01:05 And then for f times f, we have, well, the domain of f of itself.

01:09 So the domain of this is going to be f plus g, f minus g, f times g, and then f times f is going to be the same as the domain of f, which is three to infinity.

01:22 Then we look at the domain of f divided by g.

01:26 So for f divided by g, well, we have to have that g cannot be zero, but also that g must be defined for all values of g.

01:35 So that's going to be 3 to infinity.

01:46 And then for g over f, again, is going to be 3 to infinity.

01:53 But again, we do not include, yes.

02:06 So for g over f, g over f, the domain here, is going to be, again, three to infinity, but again, three cannot be included because that's going to make the denominator equal to zero.

02:19 So three to infinity.

02:21 Okay.

02:22 And then we go on to part b, and we first do here f plus g of x.

02:30 So f plus g is just what we add.

02:34 That's just the square root of x minus three plus the square root of x minus three.

02:40 So f plus g is, again, the square root.

02:43 Well, the square root of x minus 3 is f.

02:47 So the square root of x minus 3...