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) = √(x +4), g(x) = √(x − 1)

Accepted Answer

with the given functions F X equals square root of x minus seven and G X equals squared of X plus 1.5. Find F M. G. And write the domain. So let's first find what F class Diaz you will take F X plus G X. Now we look here we have two different medicals here. We will have the square x -7 plus the square root of X plus 1.5. Now they do not have the same part of the radical. So which means we can't simplify this any farther. Sorry, equation is just this here. Now I'm gonna find what the domain is of this equation to do that. We're going to check the domain of each of our functions here. So we have determinant of the square root of X -7. The final domain of a square root we're going to take was under the radical and set it equal to zero. Now we can't have a negative under the square root, which is why we're going to check this. So let's say we take our first part X -7. Set equal to zero solve for X. We would get X equals seven. On the other side will do the same for X plus 1.5. X plus 1.5 equals zero. And that makes this X equals -1.5. So the domains we can check which way we're going to be negative. So with our X equals seven let's say we let x equal six, We would have the square root of 6 -7 which is square root of -1. Since that square of a negative number. That will not work, meaning that our domain It's gonna be x greater than equal to seven or seven to infinity. Now let's check our x plus 1.5. Like here We have x equals negative 1.5. If we're gonna put a number less than 1.5 this will turn out to be negative let's say X is equal to um -3. for instance we would have screwed of negative three plus 1.5 Which that is square root of negative 1. which is also undefined meaning we also have to have X greater than equal to negative 1.5. In other words for this domain 1.5 - infinity. Now we look here we have seven to infinity and that is 1.5 to infinity. Since they're both pointing towards the same direction, they're both gonna have numbers to the right at their respective points. We're going to take the larger number because here we have 9-1.5 to infinity. But for our first time in that strictly has to be greater than seven. We're gonna take the highest point which means our domain is going to be seven to infinity. With our equation squared of x minus seven plus squared of X plus 1.5. Making this answer as answer. Okay, I hope to help you solve the problem. Thank you for watching. Goodbye.

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