find the functions f+g . f-g . f g and f / g f(x)=(x+1)/(x-1) ; g(x)=(x+2)/(x-2)

Name: Problem 23, Chapter 2: Applied Calculus for the MLSS A Brief Approach
Uploaded: 2021-06-29T14:43:33-08:00
Duration: 4 min 10 s
Channel: Manisha Sarker
Description: find the functions f+g . f-g . f g and f / g f(x)=(x+1)/(x-1) ; g(x)=(x+2)/(x-2)

step 1 In this problem, two functions F and G are given. Now, first of all, we need to determine F plus

Find the functions f+g . f-g . f g and f / g...

Key Concepts

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. They are significant in algebra because they can model complex behavior such as asymptotes and discontinuities, and they require careful attention to the values that make the denominator zero.

Operations on Rational Functions

Performing arithmetic operations on rational functions involves techniques similar to those used with numerical fractions. This includes finding a common denominator for addition and subtraction, multiplying numerators and denominators for multiplication, and multiplying by the reciprocal for division.

Simplification of Algebraic Fractions

Simplification is a key concept when working with algebraic fractions. It involves canceling common factors from the numerator and denominator to write expressions in their most reduced form, ensuring clarity and correctness in further analysis or operations.

Transcript

00:01 In this problem, two functions f and g are given.

00:04 Now, first of all, we need to determine f plus g.

00:07 Now, by the definition of the sum of two functions, f plus g of x is equal to f of x plus g of x.

00:15 Now, f of x is given to be equal to x plus 1 divided by x minus 1.

00:23 And d of x is given to be equal to x plus 2 divided by x minus 2.

00:30 So now we need to add these two terms.

00:33 The lcm will be x minus 1 times x minus 2.

00:39 And in the numerator, we will have x plus 1 times x minus 2 plus x plus 2 times x minus 1.

00:53 So the numerator will become x squared minus 2x plus x minus 2 plus x minus 2 plus x squared minus 2 plus x squared plus 2x minus 2.

01:04 And in the denominator, we have x minus 1, x minus 2.

01:08 So we can see that the plus x and the minus x will cancel out, the plus 2x and the minus 2x, these two will also cancel out.

01:17 So we have x square minus 2 and another x square minus 2.

01:22 So we have 2 times x square minus 2 divided by x minus 1, x minus 2.

01:28 And this is the required expression for f plus g of x.

01:32 Next, we need to determine f minus g.

01:39 By definition, f minus g of x will be f of x minus g of x, and this will be equal to x plus 1 by x minus 1 minus 1 minus x plus 2 divided by x minus 2.

01:57 So proceeding in a similar manner, we have in the denominator x minus 1 times x minus 2.

02:02 In the numerator, we have x plus 1 times x minus 2...

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