Hello class! Welcome back to our channel. So for today's video, I will teach you how to get the domain and range of a rational function. Okay, so we have three functions that we will solve today.
So let's start with f of x. Say for example, you are asked to find the domain and range of f of x equals 2 over x plus 1. So before we get the domain and range of our function, let's define first what domain and range means. So remember, domain is the set of all possible x values. Okay? Then range is the set of all possible y values.
Okay, so now, let's go back to f of x equals 2 over x plus 1. So, since this is a rational function, it means that our numerator has a polynomial or expression. Then, in the denominator, we have q of x. Okay, where in a rational function, if you remember our definition, our denominator should not be equal to zero. Right?
Because if our denominator will be zero, it means we will have indeterminate or undefined. So, it means that we will get the restricted values from our denominator. Okay, so again, our Q of X or our denominator should not be equal to 0. So, let's go back here to our function. We have 2 over X plus 1. So, as you can see, we have a binomial which is X plus 1. So, what are the possible values of x that our denominator will be 0? So, in our case, our values are restricted to negative 1. So, our x should not be equal or equate to negative 1 since if we put that in our denominator, we have negative 1 plus 1, the denominator of our rational function will be 0. Therefore, if we take the possible x values or the domain, we have the set of all x such that x is an element of any real numbers except negative 1. Okay, so ito ngayon yung ating domain o yung possible x values.
Nakuha ba? Then for range naman, para makuha natin si range ng ating given function, first, replace muna natin si f of x, gawin natin y. Okay?
and then solve the equation solve x so to get that we can cross multiply so we have y times x plus 1 equals 2 times y over 1 so we have 2 times 1 so that's positive 2 Correct! So, let's leave x on the left side of our equation. So, let's divide that by y to cancel this.
So, we have x plus 1 equals 2 over y. Okay, then to solve for x, we can transfer positive 1 to the right side of our equation. So we have x equals 2 over y minus 1. Okay, so now let's observe our y.
So, our y is in the denominator. So, it means, what is the value of y that is not allowed in our equation? Okay, if you remember, our denominator should not be equal to zero. So, y is not equal to zero. Okay, that is our restricted value.
Tama kasi kung ang y natin magiging 0, magkakaroon tayo ng 2 over 0 dito. So that is undefined. Tama?
So ngayon, pwede na natin... We can identify the range or the set of all possible y values. So we have the set of y or the set of all y such that y is an element of any real numbers except zero.
So this is the domain and range of our function 2 over x plus 1. Did you get it? So now let's proceed to our second function. We have x minus 2 over x plus 2. So again, to find the domain, the first step is to identify what are the restricted values of x. So let's check the denominator. We have x plus 2. So that means that our x is not allowed to be negative 2. Since when we input negative 2 here in our function, we have negative 2 plus 2 in our denominator.
And the result is 0. Correct! So if our denominator is not allowed to be 0, we need to restrict it. So what is our domain now?
So we have the set of all x such that x is an element of any real numbers except negative 2. Did you get it? So now, to find the range, again, let's replace g of x and make it y. We have y equals x minus 2 over x plus 2. Right? So y here is the same as y over 1. So we can cross multiply. So let's try.
We have y times x plus 2. equals x minus 2 times 1 that is x minus 2. Okay, so let's distribute y here in our binomial. So we have y times x that is xy plus y times 2 that is 2y equals x minus 2. Okay, so to solve for x, we need to transfer x to the left side of our equation, then the ones without x on the right side. Okay, so we have xy, our x there, when transferred, will become negative x, equals 2y, when transferred to the right side of our equation, will become negative 2y minus 2. Okay? So now, to solve for x, we need to factor out x here at xy minus x. Correct?
So let's factor out x here since that's our common monomial. So we have x times. So if we factor out x here at xy, what we'll leave here is y.
minus, we've turned out x here, so x divided by x will be 1. Right? Equals negative 2y minus 2. So now, to continue isolating x, we can divide both sides of the equation by y minus 1. Okay, so this can be cancelled, guys. So, meron tayong x equals negative 2y minus 2 all over y minus 1. Ngayon, ano kaya yung hindi pwedeng maging value ni y?
So, observe natin yung denominator. We have y minus 1. So, it means that y should not be positive 1. Since y will be 1, we will have 1 minus 1 in our denominator, which is 0. So, we need to restrict it. Now, to find the range, So we have the set of all y such that y is an element of any real numbers except positive 1. So this is the domain and range of g of x equals x minus 2 over x plus 2. Did you understand guys?
So let's proceed to... h of x so we have h of x equals x square minus 3x minus 4 all over x plus 1 so in this case okay again check natin yung denominator so we have x plus 1 sa denominator ibig sabihin x should not be equal to negative 1 otherwise magiging zero yung ating denominator So to get the domain, so we have the set of all x such that x is an element of n real numbers except negative 1. Did you get it? Now, to solve for the range, so replace h of x, let's do y equals x squared minus 3x minus 4 over x plus 1. So, if you notice guys, our numerator here is trinomial or quadratic. So, we can factor that. Right?
And, the x squared minus 3x minus 4 is factorable. Okay, so where can we factor that? So, let's try.
We have x squared here in our first term. So, it means we have x and x. Next, think of the factors of negative 4 that when combined is negative 3. So here, the factor of negative 4 possible is 4 and 1. Since negative 3 is their sum, it means that our negative here is 4, then our positive is 1. So, when we combine that, we have negative 4 plus 1, negative 3. Okay?
So, now, if you notice, we can cancel out x plus 1. So, our y, we have an equation of y equals x minus 4. Okay? So, here, if you remember, our x is not allowed to be negative 1. Okay, so if we input negative 1 to get the value of y, we can put it here. If negative 1 is the restricted value of x, so if we put that here in our equation, we have y equals negative 1 minus 4. So y, its restricted value is negative 5. Okay, so it means that if we don't have x equals negative 1, we don't have the value of y equals negative 5. Okay, so therefore, what are our possible y values?
So our range now, we have... The set of all y such that y is an element of any real numbers except wala tayong negative 5. Okay? So this now the domain and range ng ating h of x equals x squared minus 3x minus 4 all over x plus 1. Okay? So nakuha ba guys?
This is the end of our video. I hope you understood how to get the domain and range of rational functions. So if you have questions or clarifications, kindly put them on the comment section below.
Thank you guys for watching. This is Prof. D. I'll catch you on the flip side. Bye!