number one calculate the value of f of two using the function shown below so all we need to do is replace x with two so we have two to the third plus five times two squared minus nine two times two times two is eight two squared or two times two is four and five times four is twenty twenty minus 9 is 11 8 plus 11 is 19. so therefore answer choice e is the right answer number two find the inverse function using the function shown below so feel free to pause the video and work on this example so let's replace f of x with y next switch x and y and then just solve for y in order to get rid of the radical we need to raise both sides of the equation to the fifth power so that these two will cancel so right now we have x to the fifth is equal to seven y plus three i mean minus three now let's add three to both sides so x to the fifth power plus three is equal to seven y and then divide both sides by seven so x to the fifth plus three divided by seven is equal to y which is also equal to the inverse function and so this is the answer number three if f of x is equal to negative 15 what is the value of x so let's replace f of x with negative 15. now since we have a negative in front of 2x squared let's take everything from the right side and move it to the left side so if we have a negative in front of the 2x squared on the right side on the left side is going to be positive 2x squared 5x is positive on the right side it's going to be negative on the left negative 3 is going to change to plus 3. and let's keep the negative 15 on the left side 3 minus 15 is negative 12. so now we need to factor we have a trinomial where the leading coefficient is not one so we need to multiply the leading coefficient by the constant term in order to factor it 2 times negative 12 is negative 24 and two numbers that multiply to negative 24 but add to the middle coefficient negative five that's going to be negative eight and three so now let's replace the middle term that is negative five x with negative eight x plus 3x and then we could factor by grouping so let's take out the gcf in the first two terms and that's going to be 2x 2x squared divided by 2x is x negative 8x divided by two x is negative four and now in the last two terms let's take out a three and so this is what we're gonna have now we can factor the gcf which is x minus 4. if we take that out we'll be left with 2x plus 3. now let's set x minus four equal to zero and two x plus three equal to zero so x is four two x is negative three and x is negative three over two so we have two answers but the only one that's listed is answer choice d number four what is f of g of x so what do we need to do here first let's replace g of x with what it's equal to that is x minus five so we're looking for f of x minus five which means that we need to take x minus five and plug it into each x value that we see here so it's x minus five squared plus three times x minus five plus two x minus 5 squared that's x minus 5 times x minus 5. if we foil it it's going to be x squared minus ten x plus twenty five if we distribute three to x minus five it's going to be three x minus fifteen and then plus two so now let's combine like terms we can add these two and then we can combine those negative 10x plus 3x is negative 7x 25 minus fifteen is ten ten plus two is twelve so the answer is x squared minus seven x plus twelve number five what is the value of g of f of two so first let's find the value of f two let's replace x with two two to the third is equal to eight five times two is ten and eight minus ten is negative two negative two plus one is negative one so that's the value of f of two and we're looking for g of f of two and since f of 2 is negative 1 we need to find g of negative 1. so now let's use this equation so it's 7 minus negative 1 squared negative 1 squared is negative 1 times negative one that's positive one and seven minus one is six so g of f of two is equal to six therefore b is the right answer number six are f and x and g of x are they inverses of each other how can we prove if they're inverses of each other so we have to show that f of g of x is equal to x and also g of f of x we need to prove that this is equal to x if we can do that that we know for certain that f and x and g of x are inverses of each other so let's start with the first one f of g of x let's plug in g of x into f so this is going to be f of one third x squared minus one so we're going to take this and insert it into this equation so it's going to be the square root of 3 and then instead of writing 3x we're going to replace x with 1 3 times x squared minus 1. then we have a plus one inside the square root three times one third is simply one so we have one times x squared minus one and that's just x squared minus one negative one plus one adds up to zero and the square root of x squared is x now let's move on to the other side let's replace f of x with the square root of three x plus one and let's plug this into this equation so it's going to be one third and instead of writing x squared we're gonna write square root three x plus one squared minus one so the square and the square root will cancel and so we're just going to get three x plus one and then we have a minus one outside of that one minus one adds up to zero so what we have left over is one third of three x which is equal to x therefore f of x and g of x are inverses of each other number seven which of the following graphs is not a function in order to find the answer we need to see which one does not pass the vertical line test starting with a if we draw a vertical line we can see that it touches it only once which means that a is a function if we do the same thing for b it touches the vertical line only once so b is a function and d is also a function it passes the vertical line test but c is not a function if we draw it at that location we can see that it touches the vertical line three times so the right answer is c number eight which of the following is not a one-to-one function is it a b c or d in order to find out which one is not a one-to-one function we need to see which one will fail the horizontal line test so starting with a if we draw a horizontal line we can see that the curve touches it once so a is a one-to-one function it passes the horizontal line test if a function is one-to-one that means that the inverse function is a function which means the inverse function passes the vertical line test so looking at b if we draw a horizontal line it passes the test it touches the point only once so b is a one-to-one function and the same is true for d d is one to one as well looking at c it does not pass the horizontal line test it touches the horizontal line at two points so therefore c is the answer c is not a one to one function if you were to draw the inverse function for c it will look something like this and since c is not a one-to-one function since c doesn't pass the horizontal line test the inverse will not pass the vertical line test and so the inverse function is not a function number nine if f of four comma y is 48 what is the value of y so we can replace this with 48 because the function is 48 and x is equal to four the first number corresponds to x the second number corresponds to y and our goal is to find the value of y four squared is four times four that's sixteen three times four is twelve and so this is what we have now we could divide everything by two because all the numbers that we see here are even so half of 48 is 24 half of 16 is 8 half of 12 is 6 half of 2 is 1. and now let's subtract both sides by 24. let's write it in standard form so it's y squared plus 6y and 8 minus 24 is negative 16. so let's go ahead and factor what two numbers multiply to negative 16 but adds a positive six so this is eight and negative two so this is going to be y plus eight and y minus two so therefore y is equal to negative eight and it's equal to 2. 2 is the answer because that's the one that's listed so b is the right answer number 10 what is the domain of f divided by g so f divided by g that's going to be seven x minus three divided by x squared plus two x minus fifteen so x can be anything except any value that's going to produce a zero in the denominator so therefore x squared plus 2x minus 15 cannot equal zero let's go ahead and factor the expression what two numbers multiply to negative 15 but add to the middle coefficient too this is positive five and negative three so it's going to be x plus five times x minus three so therefore x plus five cannot equal zero and x minus three cannot equal zero so x cannot equal negative five and x cannot equal three so x is all real numbers except those two values so to write the answer using interval notation it's going to look like this negative infinity to negative five that means x could be anything from negative infinity to negative five but it doesn't include negative five union x could be anything between negative five and three but not negative five or three and then union three to infinity x can be anything greater than three so that's the domain of f divided by g