For the indicated finctions fand g. find the functions f+g, f-g, fg, and flg, and find their dom

For the indicated finctions fand g. find the functions f+g,...

Key Concepts

Pointwise Operations on Functions

Pointwise operations refer to the process of defining new functions by performing arithmetic operations on the outputs of functions at each value in their domains. This includes addition, subtraction, multiplication, and division, where the resulting function at any input x is determined by applying the operation to f(x) and g(x).

Function Addition and Subtraction

When adding or subtracting functions, the operations are performed for each input value separately. For example, (f+g)(x) is computed as f(x) + g(x), and (f-g)(x) as f(x) - g(x). The domain of the sum or difference is the set of all x for which both f(x) and g(x) are defined.

Function Multiplication

Multiplying functions involves taking the product of the values of the functions at each input. In other words, (fg)(x) is found by multiplying f(x) by g(x) for every x in the intersection of their domains. The domain of the product is similarly the common domain where both original functions are defined.

Function Division and Domain Restrictions

Division of functions is defined as (f/g)(x) = f(x) / g(x), and it is crucial to note that the divisor g(x) must not be zero. Thus, the domain of the quotient includes only those x-values for which both f(x) and g(x) are defined, and in addition, g(x) ? 0.

Domain of Polynomial Functions

Polynomial functions are defined for all real numbers because they do not have any operations like division by a variable or square roots of negative numbers. However, when combined with other functions in operations like division, attention must be given to any additional restrictions imposed by the operation.

Transcript

00:01 So in this problem, we will take these functions f and g, and we will find f plus g, f minus g, f times g, and f divided by g.

00:11 We'll also find the domains of those functions.

00:14 So let's begin by finding f plus g of x.

00:20 So we're adding together 2x minus 7 and 9 minus x squared.

00:28 So generally, we like to write our powers of x in descending order, meaning that we'll start with the highest power negative x squared, and then we have plus 2x, and then negative 7 plus 9 is 2.

00:46 Writing it in a different order is totally okay, but this is typical.

00:52 So this is our function.

00:54 And then this is a quadratic, so we can plug in anything we like for x.

00:59 There are no restrictions, so our domain is negative infinity to infinity.

01:05 Next we'll do f minus g of x.

01:13 So 2x minus 7 minus 9 minus x squared.

01:20 So our x squared term becomes positive.

01:24 We still have our 2x.

01:27 And now we have negative 7 minus 9.

01:30 So negative 16 as our constant term.

01:35 So again, this parabola in terms of x has no restrictions for the domain.

01:40 So negative infinity to infinity is our domain.