The green dotted curve below is a graph of the function f(x)f\left(x\right)f(x). Find the domain and range of g(x)g\left(x\right)g(x) (the blue solid curve), which is a transformation of f(x)f\left(x\right)f(x).

Dom: [ 1 , 5 ] \left[1,5\right] [ 1 , 5 ] , Ran: [ − 5 , 1 ] \left[-5,1\right] [ − 5 , 1 ]

The green dotted curve below is a graph of the function f ( x ) f\left(x\right) f ( x ) . Find the domain and range of g ( x ) g\left(x\right) g ( x ) (the blue solid curve), which is a transformation of f ( x ) f\left(x\right) f ( x ) .

Hey, everyone, and welcome back. So let's see if we can solve this problem right here. In this problem, it is said that the green dotted curve that we see below is a graph of the function f of x, and we're asked to find the domain and range of g of x, which is the blue solid curve given that this is a transformation of f of x. Now to solve this problem, what I'm first going to do is try to recognize how this curve is transformed. And it doesn't look like this curve has been stretched in any way because we don't see any kind of squeeze happening to the curve, and it does not look like the curve has been reflected in any way either. But I can see that the curve does appear to be in a new location, so it appears we are dealing with a shift transformation in this graph. Now if I go ahead and just look at a random arbitrary point on our function, our initial function f of x, before we went through the transformation, I can see here that it appears that we have been shifted 1, 2 units to the right, and 1, 2, 3 units down. So this is how the function got transformed from the original f of x to g of x. Now what I can also do is take a look at the domain and range of the original function, which is given to us. We can see that the domain of our original function, and I'll write it over here, goes from negative 1 to 3, and we can see that the range of our original function goes from negative 2 to 4. So based off this, what I can do is look at this transformation and figure out how this domain is going to change. Because the graph got shifted 1 to 2 units to the right, that means that the domain, which refers to the allowed x values, is going to change by 2. So the domain of our transformation g of x is going to be and I'll actually write this for g of x, is going to be that our domain goes from well, if we add 2 to negative 1, we would have positive one to go negative 1, 0, then 1, so we'd have positive 1. And if we add 2 to 3, we would get 5. And so this is going to be the domain of our transform function. We can also see here for the range that and by the way, this is the domain up here. So we can see for the range that our function is has been shifted down 3 units, so what I can do is take away 3 from this 2. If you take away 3 from this 2, we're going to get negative 5 because negative 2 minus 3 would give you negative 5. And if I take away 3 from this 4, we would get 1. So this is how the domain and range is going to behave for our new shifted graph, g of x. Now if you wanna prove this to yourself, you can also use the squish method we've discussed in previous videos. So what I can do is take this curve that we see here, and I can squish it down to the x axis. If I do this, we're going to get a line that goes from positive one to positive five, and that's the domain of this graph, so this is clearly the right domain. And if I go ahead and squish this to the y axis, that will give me the range. So if I squish this down to the y, we're going to get a line that goes from negative 5 all the way up here to positive 1, and we do need to include both these values since we have 2 solid dots or solid curves. And we can see here that we go from negative 5 to positive 1, including both values. So this range was clearly correct as well. So this is how you can find the domain and range of a shifted or transformed graph. Hope you found this helpful.