Given the functions f and g, find f+g, f-g, f ·g, and (f)/(g), and state the domain of each.

Given the functions f and g, find f+g, f-g, f ·g, and...

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Key Concepts

Function Operations

This concept deals with combining functions using arithmetic operations such as addition, subtraction, multiplication, and division. When functions are operated on, the resulting function's domain is typically the intersection of the domains of the original functions while also taking into account any further restrictions introduced by the operation, such as the need to avoid division by zero.

Domain of a Function

The domain of a function is the collection of all input values for which the function is defined. When determining the domain, one must consider any restrictions imposed by the function’s formula, such as avoiding negative values under an even root or excluding values that cause a zero denominator in a fraction.

Square Root Functions

Functions involving a square root require that the radicand (the expression inside the square root) is nonnegative in order to produce real numbers. This restriction determines part of the domain and ensures that the output is a real number.

Rational Functions

Rational functions are those expressed as a quotient of two expressions, typically polynomials. The key restriction for rational functions is that the denominator must not be zero, as this would lead to an undefined expression. Identifying and excluding these values is essential when determining the domain.

Transcript

00:01 We first want to find the sum function f plus g, which is f of x plus g of x.

00:06 And in our case, that would be the square root of 1 minus 2x plus 1 over x.

00:17 If you need to write this as a single fraction, you could.

00:23 The common denominator would be x.

00:26 So we would need to scale the numerator by that as well.

00:28 And it would be x square root of 1 minus 2x plus 1 all over x.

00:35 Or you could have just left it in the original.

00:38 I'm sure that would have been fine too.

00:40 Now let's think about the domain.

00:42 It's working interesting here.

00:47 Let's start with the radical portion.

00:49 We wouldn't want or we would want what's under the radical to be greater than or equal to zero because that's the only time we can square root numbers.

00:59 And so negative 2x is greater than equal to negative.

01:02 1 divide by negative 2 and we switch the sign x is less than or equal to positive 1 half we also can't have a denominator of 0 so x cannot be 0 so in interval notation we would say that this is going to go from negative infinity up to 0 not included union 0, like infinity, i think.

01:45 0 to 1 half included.

01:50 Again, negative infinity to 0, union 0 to 1 half included.

02:09 Do the difference function now...

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