In Exercises 61–64, find the domain of each function. f(x) = √...

Question

In Exercises 61–64, find the domain of each function. f(x) = √(2x/(x + 1) - 1)

Accepted Answer

Hey everyone here we are asked to find the domain of the function which is the square root of five X divided by the quantity of X plus four minus four. Now we want to start by finding where X is undefined. So we want to look at this denominator here and note that when you divide by zero, you return in undefined result. So we can set this denominator of X plus four equal to zero to find where X is undefined. So just subtracting four from both sides, we can see that X is equal to negative four and therefore negative four cannot be included within the domain because it will create an undefined function. And now we can look at this entire expression underneath the square root and note that it must be positive because if it was negative we would then be taking the square root of a negative which is undefined. So using this we see that her expression five X divided by the quantity of X plus four minus four must be greater than or equal to zero. And how using these two pieces of information, we can start to write the domain which we found so far. So we can see that X is within negative infinity to negative four. And this is because negative four is on the left side of the equation. So all negative values before it can be inputs as well. And we want to note that this infinity sign is always closed by a parenthesis because it is always considered an open interval and this negative four is also closed by a parenthesis because it is undefined at this value and therefore cannot be equal to negative four. And now from here we can start solving for X in our inequality to find more information about our domain. So we can start by adding four to both sides, which gives us five X divided by the quantity of X plus four is greater than or equal to four. And now we can multiply both sides by X plus four to get rid of the denominator, which gives us five, X is greater than or equal to four X plus 16. And now to get X all on one side will subtract four X from both sides giving us X is greater than or equal to 16. And now with this inequality we can rewrite it as a domain and interval notation and it will be X is within 16 as the lower bound to infinity because X can be greater than or equal to 16 and because it can be equal to 16, there is a bracket here which closes it. And again, the the infinity sign is closed by a parentheses. So now we can combine the two intervals we have found which gives us X is within negative infinity to negative four and 16 And infinity. And since these intervals do not overlap, this is our final answer here, which is choice B where the domain is negative infinity to negative four and 16 to infinity. Thanks for watching, I hope you found this video helpful, and I'll see you again next time.

In Exercises 61–64, find the domain of each function. f(x) = √(2x/(x + 1) - 1)

Key Concepts

Domain of a Function

Square Root Function Restrictions

Rational Expressions and Undefined Points

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