Hi guys! Welcome back to our channel. So for today's video, tuturuan ko kayo kung paano ba kunin yung sum, difference, product, and quotient of two functions. So ang title ng video natin is Operations on Functions.
So usually, lumalabas to sa grade 11 under General Mathematics. So simulan na natin. So dito guys, pag pinapakuha yung f plus g of x, same lang yan ni f of x. plus g of x.
Okay? Then, kung pinapakuha naman yung difference between the two functions o yung f minus g of x, same lang yan ni f of x minus g of x. Okay?
Then, for multiplication naman, if we have f times g of x, same lang yan ni f of x times g of x. Okay? And last one, we have the quotient of two function, f of...
F over g of x, same lang yan ni f of x over g of x. Okay, kung saan yung g of x dapat hindi zero para hindi tayo magkaroon ng undefined. Okay, so let's have our example. So given the two function f of x equals x square minus 1 and g of x equals x square minus x. So compute the sum, difference, product.
and quotient of the two functions. Okay? So, unayin natin yung sum. We have f plus g of x. So, again, kagaya ng pinakita ko kanina, same lang yan ni f of x plus g of x.
Okay? So, kung saan, ang f of x natin is equal to x squared minus 1 plus Ang g of x naman natin is equal to x square minus x. Okay?
So dito guys, paano ba tayo mag-plus ng polynomial? So dito, pagka mag-a-add kayo, make sure na mag-a-add kayo ng similar terms. Okay?
Ang x square para kay x square, yung x natin para kay x, then constant for constant. Nakuha ba? So unayin natin yung x square.
We have here x squared plus x squared. So that is 2 then copy x squared. Nakuha?
Next, dun naman tayo sa x variable. Kung mapansin nyo, isa lang yung ating x variable which is negative x. So bring down lang natin yan.
Then sa constant naman, isa lang din which is negative 1. So bring down lang. Nakuha ba? So, kung wala na kayong makombine, so that will be our f plus g of x.
Okay, so final answer. Nakuha ba? Next, dun tayo sa difference. Yung f minus g of x. So, sa pinakita ko kanina, same lang yan guys ni f of x minus g of x.
Okay? Kung saan? Ang ating f of x is equal to x squared minus 1. Ang g of x naman natin is equal to x squared minus x. Okay, so para lang siyang kagaya nung kanina na addition.
Yun nga lang, dito minus yung ating operation. So paano ba tayo mag-subtract ng polynomials? So ang ginagawa natin dyan is bring down lang natin yung first term natin.
And then, i-change natin yung ating operation. From subtraction, magiging addition. Then, pag ginawa na natin yung steps na yan, make sure na papalitan natin yung sign ng mga terms dito. So, yung x squared natin magiging negative x squared. Then, yung negative x natin magiging plus x.
Nakuha ba? So, in-apply natin yung rule ng subtraction. Okay, then combine similar terms.
So, pag nag-combine kayo dyan, we have, so, x squared plus negative x squared. So, maka-cancel na yan. Tama? And yung x naman natin, wala naman siyang similar term, so bring down.
And then yung negative 1, wala rin similar term, so bring down. So, we have x minus 1, and that will be our f minus g. Of x. Okay, so this will be our final answer.
Nakuha ba? Next, let's proceed to multiplication. We have f times g of x. So, same lang yan guys ni f of x times g of x.
Okay? Kung saan, ang f of x natin is equal to x square minus 1. Then, ang ating g of x naman, we have x square minus x. Okay?
So, paano ba tayo mag-multiply ng binomials? So, dito ang pinaka-common na ginagamit natin is yung FOIL method. Tama?
So, kung saan, minumultiply natin yung first terms. So, x squared times x squared, that is x raised to 4. Diba? Pagka nag-multiply tayo, ina-add natin yung exponents. Then, for outer, we have x squared times negative x. So, that is...
Negative x cubed. Okay? Sa inner naman, we have negative 1 times x squared.
So that is negative x squared. And for our last term, we have negative 1 times negative x. That is positive x. Okay?
So may nakikita ba kayong similar terms? So since wala naman, so ito na yung ating final answer. We have... f times g of x. Okay?
Final answer. Nakuha ba? So, let's proceed to our last example. We have f over g of x.
So, same lang yan, guys. Ni f of x over g of x. Tama?
Kung saan, ang f of x natin, we have x square minus 1. over g of x which is x square minus x. Okay? So paano ba tayo nagsisimplify ng rational expression?
So dito, ang common technique natin is pina-factor natin yung ating numerator and denominator. Kukunin natin yung factor. Okay? So ano bang factor ng x square minus 1?
So that is... The product of sum and difference, diba? Kung mapansin nyo, perfect square yung first term, x square.
So we have x and x. Then yung 1, that is 1 times 1. Tama? So sum and difference, so that is plus minus. Nakuha ba? Over.
We have x square minus x sa denominator. Kung mapansin nyo, meron tayong common monomial. Ang common monomial natin is x, so kung saan, pwede natin siyang i-factor out. Okay? So paano natin makukuha yung nasa loob ng ating parentheses?
We have x squared divided by x, yung finactor natin, so that is x, minus x divided by x, so we have 1. Tama? So sino yung common? So mga makikita nyo, Nakomon si x minus 1 sa numerator and denominator.
Tama? So, yung may iiwan natin, that will be our f over g of x. So, ano yung naiwan?
We have x plus 1 sa numerator over x sa denominator. Okay, so this will be our final answer. Nakuha ba?
So, this is the end of our video guys. So, if you have questions, pakicomment na lang sa ating comment section below. Okay? So, thank you guys for watching.
This is Prof. D. See you on the flip side. Bye!