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) = 2x² − x − 3, g (x) = x + 1

Accepted Answer

for the given functions F X equals five X squared plus 33 X minus and G X equals X plus seven. Find F divided by G. And write the domain. Let's first find what F divided by G is so F over G five x squared plus 33 x -14 Divided by X-plus seven. So our first step here is to try and factor out our five X squared plus 33 X minus 14 because we can actually simplify this problem. So let's shine factor of this polynomial. So first thing we can look at is our five X squared. Yeah, private squared. The factors of five X squared are just five X and one X. Because five only has one factor. So we have five x and X. Now we look at our 14. So you find two numbers that multiply together to equal -14 and also some to equal 33. We can make use of here. It's actually the A. C method. So we say a C. You take any time see And find the factors of that. So we have 14 tons five, That's gonna be 70. Now we want to find the factors of 70. So Look here, you would have won 72 and 35. 05.14.7 and Sniff Real. Which two of these numbers are going to add up to equal a 33. Remember one of these are also negative because we have a negative in front of the 14. Let's go ahead and say we have R two and R 35. We noticed there if we have a negative two and positive 35 we will add up to a positive 33. So I rewrite this. We have five X squared. Let's go ahead and say put our 35 1st plus -2 X -14. Now we can factor out his first two terms and his last two terms. So we have five X squared plus 35 X. Well five is uncommon with both of these terms. So check out five. Give us X squared plus seven X. X. is also common between these two terms. That gives us five x Times X-plus seven. Now for native two X minus 14 negative two is actually common between both terms, expect out a negative two now negative two X plus seven. So if you notice here we have five x minus two. On the outside X plus seven on the inside of these parentheses we can rewrite this with the stuff on the outside group together. Five x minus two and stuff on the inside group together. X plus seven. This is five X plus two side minus two and X plus seven. So if we go back to our function you can say it F. O. R. G sequel to 5 6 months to export seven Over X-plus seven. Now if you notice one more thing here we have an expo seven on top expo seven on the bottom those can cancel out, cancel those out. We get f over G equals five X -2. Now I want to find the domain of this function. Since at the start of the function we had a fraction. We're gonna go ahead and take the domain of the suspect to that function. So you look here have an X plus seven and the denominator to determine the restriction for this. You want to see what X. Value makes the whole denominator equal to zero. So let's go ahead and take X-us seven equal to zero. We solve for X. Track seven. We get x equals -7. So we have our domain the all real numbers except X equal to negative seven. We can rewrite this as negative infinity to negative seven And -7 to Infinity. That is our domain With that equation being five X -2. Where she was the answer to this problem is answer D. Okay I hope that you solve the problem. Thank you for watching. Goodbye

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In Exercises 31–50, find ƒ/g and determine the domain for each function. f(x) = 2x² − x − 3, g (x) = x + 1

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