In Exercises 31–50, find fg and determine the domain for each fun...

Question

In Exercises 31–50, find fg and determine the domain for each function. f(x) = √x, g(x) = x − 4

Accepted Answer

for the given functions F X equals square to three X. And G X equals six X plus seven. Find F times G. And write the domain. Let's go ahead and find what F times G. Is. So F times G. This is equal to our two equations, multiplied by each other. So we get squared of three X. Multiplied with six x plus set. Now we can actually simplify this a little bit more and that X. Is also underneath the radical there. But we can go ahead and multiply the ship. So I would just shoot my squared of three X. To the sixth X. And to the seven. So if we read that we will get six x. 3 x Plus seven Squares of three X. So now we can actually revise this even further. We notice here we have a six X squared of three X. One of our expanded rules that we should know states that if we have square root something like three X. We can rewrite those as a product of square roots. Or in other words we can write this square 23 times a squared of X. So if we go here, we write this we have six squares of three times X squared of X. And then the 7 sq3 eggs will remain the same. Now we can look at another exponent role that we have. We first need to look at the squared of X. One thing we should know also is that the square root of X is equal to X to the one half power. Now we're multiplying two exponents together here we have an X. That's technically saying X. To the first I read that there to the first. So we actually have X. To the first times X. To the one half. When you multiply exponents together, you just add the exponents and the exponent area. So in other words this is X to the power of one plus one half. We can simplify this even further by saying X to the three halves. So if I go back over to our problem we will rewrite this as six squares of three Times X. The 3/2ves and the plus seven squared of three X. Will remain the same because you can't simplify it anymore. So if you look here we cannot simplify our problem anymore. So this would be our equation F times G. Now we want to find the domain of this equation to find the domain we want to look at see if we have anything in our function that could be can give us a problem. In other words, our square root function. Let's take a look at that real quick. I wanna see if anything will give us a restriction. So I care. We have X. 23 halves and squared of three X. Let's go in and take the squared of three X. And see what the domain of this is. So the square root of three eggs. I'm gonna be here. We have a square root. We want to see what number makes a square root value negative underneath the radical because we can't have a negative square root. It's just not possible. So our to rewrite this let's go and take what we have underneath the square root and set it to 03 X. Equals zero. And we'll solve for X. five x 3 X equals zero. So zero is the boundary point for our domain are left bound and this case is going to be zero because you can't have anything negative zero to infinity. Now let's take a look at X. The three halves. Well Extra 3/2. Thera Picks 3/2. Extra 3/2. We can rewrite this just to be the square of X to the third. Which in this case is quotes can still not be negative because we have a score of X. Which means this domain is also zero to infinity because both domains are zero to infinity. Our domain overall will be zero to infinity. With the equation six squares of three times exit three halves plus seven times X squared of three X. The answer to our equation is answer B. Okay I hope that will solve the problem. Thank you for watching. Goodbye

In Exercises 31–50, find fg and determine the domain for each function. f(x) = √x, g(x) = x − 4

Key Concepts

Function Composition

Domain of a Function

Square Root Function

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