In Exercises 61–64, find the domain of each function.f(x) = 1/√(4...

Question

In Exercises 61–64, find the domain of each function.f(x) = 1/√(4x^2 - 9x + 2)

Accepted Answer

Hey everyone here we are asked to find the domain of the function F. Of X. Which is equal to six divided by the square root of three, X squared plus 10 X minus eight. Now the first thing we want to do is note this expression in the denominator and factor it out. So we get six, divide by the square root of the quantity of three X minus two times the quantity of X plus four. And now from here we need to note that this expression is below in a square root so it must be positive since taking the square root of a negative is undefined. Additionally, this expression is a denominator so it cannot be equal to zero since dividing by zero is also undefined. So using this information we can write the expression as an inequality. So we have the quantity of three X minus two times the quantity of X plus four must be greater than zero. And from here we can find values of X which we can use to find the domain. So we can find values of X by cell. Each set of parentheses equal to zero. So we have three, X minus two is equal to zero, adding two to both sides. We have three. X is equal to two. Getting X alone. We divide both sides by three. Giving us X is equal to two thirds repeating the same idea for X plus four. We set it equal to zero, subtracting four from both sides. We get X is equal to negative four. And now from here we can take to these two values and place them on the number line here and compare them to find the domain. So for each value they will have an open point because they aren't included within the domain since they cause the expression to be equal to zero. So next we need to note that going right, we go to positive infinity and going left, we go to negative infinity. And so next we can start testing in between and outside the two values. So starting with in between, let's say we test zero while zero gives us a negative times a positive resulting in a negative value. Next, if we test a number less than negative four, let's say negative six, we get a negative times a negative resulting in a positive. Next, if we test a number larger than two thirds, let's say two. We get a positive times a positive resulting in a positive. So from here we can see that we go from negative forward to negative infinity because that is within the inequality as we decided above. And it can go from two thirds to positive infinity. So next we can write this information as to inequality. So we have X is less than negative four or X is greater than two thirds. And using these two inequalities, we can use the integral notation to write the domain. So we have X is within negative infinity to negative four and two thirds to infinity. So here it is important to note that both infinities are closed by parentheses, and the negative four and two thirds are also closed by parentheses, since they are not included within the domain. So that leaves us with answer choice A with a domain of negative infinity to negative four and two thirds 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) = 1/√(4x^2 - 9x + 2)

Key Concepts

Domain of a Function

Quadratic Functions

Inequalities

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