Today we'll be talking about the algebra of functions. The goal is to find the sum, difference, product, and quotient of two functions. If we let f and g be functions and x is in the domain of both, f plus g of x is equal to f of x plus g of x.
f minus g of x is equal to f of x minus g of x. f times g of x is equal to f of x times g of x. And f divided by g of x is equal to f of x divided by g of x.
So let's take a look at how we are going to use this. If we're given two functions, we should be able to find the sum, difference, product, and quotient of the functions at a specific value of x. So by applying these definitions, we know that f plus g of negative two is equal to f of negative two.
plus g of negative two. So now we just have to perform our substitution into each function and then find the sum. To find f of negative two, we have to replace x with negative two in function f.
That would look like this, plus g of negative two, which would look like this. Simplifying each of these, here we'd have eight plus three, or 11. And here we'd have negative two squared, which is four minus three, which is equal to one. So, f plus g of negative two is equal to 12. The point is not really to do a bunch of number crunching, so let's go ahead and use the graphing calculator to assist us on these next three. So what we can do is type f into y one and g into y two. Now let's make sure our table is set on ask.
So let's press second window, scroll down to independent variable, highlight ask and hit enter. Now let's go to the t table. Second graph. So for example on this first problem if we look at the values of the functions at x equals negative two. If we press negative two we can see that y one plus y two would equal twelve which is what we found for the answer.
for this first problem. Now going back to our examples, f minus g of five is equal to, here are our rules, f of five minus g of five. Well let's go back to our graphing calculator and find these values.
So if we press five enter, remember that y one is f and y two is g. So here we'd have negative 17 minus 22. which is equal to negative 39. That would be f minus g of five. On this next problem, f times g of zero is equal to f of zero times g of zero.
Well this one we can probably do without the graphing calculator. F of zero would be three, g of zero, zero squared minus three would be negative three. Well three times negative three of course is negative nine. F times g of zero is equal to negative nine. And the last problem, f divided by g of two is equal to f of two divided by g of two.
So if we go to the graphing calculator, press two for x, we would have f of two equal to negative five, g of two equal to positive one. So negative five over one of course is equal to negative five. F divided by g of two is equal to negative five. Now another way to approach the same idea is by looking at it graphically.
So if we first apply the definition f plus g of eight is equal to f of eight plus g of eight. Remember these are function values therefore we're talking about y values. F of eight, here's our function F, here's X equals eight.
F of eight would equal positive three. G of eight would equal negative four. So our sum would equal negative one.
On the next problem, F times G of negative four is equal to F of negative four times g of negative four. Let's find these function values from the graph. F of negative four, here's F, here's x equals negative four. The y value or the function value is equal to zero times g of negative four, g is in red.
G of negative four is equal to positive two, which of course would give us zero. And on this last example, we do not have enough information to answer this question because we do not have the equations for the functions. Thank you for watching and have a good day.