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

Question

In Exercises 61–64, find the domain of each function.f(x) = √(2x^2 - 5x + 2)

Accepted Answer

Hey everyone here we are asked to find the domain of the function F of X. Which is equal to the square root of three, X squared plus 10 x minus eight. Now the first step here is to factor out this expression that's within the square root. So we get the square root of the quantity of three x minus two times equality of X plus four. And from here we need to note that this expression is within this square root so it must be positive because we can't take the square root of a negative. So from here we can use this information and rewrite this expression as an inequality. So we have the equality of three x minus two times. Equality of X plus four must be greater than or equal to zero. We need to not forget that we can take the square root of zero. So we need to include that in the inequality. So from here we can set both sets of parentheses equal to zero to find values for X which we can use to find the domain. So we have three X minus two is equal to zero, adding two to both sides. We have three, X is equal to two and finally dividing both sides by three. We have X is equal to two thirds and next repeating four X plus four. We set it equal to zero, subtract four from both sides. We get x equal to negative four and next we can take these two values and plot this on a number line to compare them and we need to note that going right, we go to positive infinity and going left. We go to negative infinity. Next we also need to note that both values get a solid dot because they are included within the domain. And next we can start testing in between and outside the values. So starting with a value less than negative four, we could say negative seven. Well negative seven gives us a negative times a negative, returning a positive value. Testing in between we can say we can test zero. Well zero gives us a negative times a positive resulting in a negative. Next testing a value greater than two thirds. Let's say 33 gives us a positive times a positive resulting in a positive. And so from here we can see since we need to be greater than or equal to zero, we can go from negative four to negative infinity and two thirds to positive infinity. So writing these as this information. As to inequalities we have X can be less than or equal to negative four and X can be greater than or equal to two thirds. And now we can use this information and write our domain in integral notation. So we have X is within negative infinity 2 -4 which is closed by a bracket since it is included and two thirds which is also included and enclosed in a bracket to positive infinity. Also, both infinities are always enclosed by parentheses. So now we are just left with that answer, which is answer choice C, which is a domain from negative infinity to negative four and two thirds to positive 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^2 - 5x + 2)

Key Concepts

Domain of a Function

Square Root Function

Quadratic Inequalities

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