In Exercises 31–50, find ƒ+g and determine the domain for each fu...

Question

In Exercises 31–50, find ƒ+g and determine the domain for each function. f(x) = 2 + 1/x, g(x) = 1/x

Accepted Answer

Welcome back. I'm so glad you're here. We have got two functions F of X and G of X. And we are to find F plus G and we are to write the domain. So for F plus G we're going to take our F of X and add it to our G of X. R f of X is 12 plus 3/2 X. And we're going to add that to our G of X which is 3/2 X. Well, what terms can we add together? These two fractions with two X in the denominator? Those are like terms, we can add them together. And these fractions already have a common denominator, which is great. So we can just add a cross for the numerator will bring down 12 plus. So adding across for R D numerator, excuse me for our enumerators, three plus three equals six divided by And our common denominator is two X. So we have two X. We can simplify that. 6/2. X. Six divided by two. Well, that's going to give us a three in the numerator and a one in the denominator. So one times X is still X. Bring down our 12 plus. That's as simplified as this one is going to get so 12 plus three over X. That is F plus G. Now for the domain for our domain we want to look at our original functions of X and G of X. And we want to see if there are any potential domain restrictions. And yeah, we've got fractions. So we do have domain restrictions and those fractions have an X in the denominator we can't have a denominator equal to zero. So we have to figure out what values for X would make our denominators equal to zero. And therefore undefined. So what we do is we set our denominators equal to 02, X equals zero. They're the same denominator for each function. So we only have to do this once to X equals zero, will divide both sides by two. And that means that if X equals zero, we would have a zero in our denominators and our functions would be undefined. How do we express an interval notation that X can be any number except for zero. We start with the parentheses and negative infinity, meaning starting at negative infinity. All the way up to zero. X. Could have been any of those negative numbers, but it can't be zero. So we put a parentheses to say it can't be zero. That's in union with another parentheses to say it can be any number just after zero. All the way up to. And there's no more restrictions. So all the way up to positive infinity. We close infinity with the parentheses to say that we can never quite actually obtain infinity. We look at our answer choices and that matches with answer choice. A Well done. We'll catch you on the next one

In Exercises 31–50, find ƒ+g and determine the domain for each function. f(x) = 2 + 1/x, g(x) = 1/x

Key Concepts

Function Addition

Domain of a Function

Rational Functions

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