Graph each function. Give the domain and range. See Example 3. ƒ(...

Question

Graph each function. Give the domain and range. See Example 3. ƒ(x) = 2^(x+2) - 4

Accepted Answer

Hello everybody. I hope you're doing. All right. Today, today we're going to be looking at this math question that states provide the domain and range of the function by graphing it. And the function they give us is up of X is equal to 11 to the exponent X plus nine and outside of the exponent ele uh minus 19. So when looking at a question like this, the first thing I want to identify is the parent function. So in this case, our parent function will be 11 to the exponent X. So that is the parent function here with no added uh transformations. So now we have to see what the graph for that would look like. So let's just draw a simple graph for that. Now that graph would have a Y intercept uh zero comma one and it would extend upwards past that and have a point at one comma 11. And it'll also go downwards and approach the X axis but never touch it with a point at negative one comma one divided by 11. So this general graph would have a domain between negative infinity and positive infinity since domain has to do with the possible X values here. And our graphic extending infinitely to the left and infinitely to the right and arrange the range. On the other hand, has to do with the possible Y values here. Our graph is approaching zero or approaching the X axis but never touching it. So we'll have an open bracket of N N zero, comma an open bracket because we never touch zero. So we'll have zero comma and then we're extending the graph infinitely upwards. So it'll be zero, comma positive infinity. So now, once we have that general graph of the parent function, we have to see what transformations we're doing to the parent function and how that's gonna change it. So the first transformation we're gonna add is 11 X and then now we're adding plus nine to the exponent. This plus nine means that we're shifting the graph of 11 X nine units left. And then we also have the transformation of 11 X plus 11 to the exponent X plus nine. And then outside of the exponent, we have a minus 19, which means we are shifting the graph of 11 to the exponent X plus nine, 19 units down. So let's now try some points for reference, if X is equal to negative nine, Y is equal to one minus 19 because if you plug in nine or negative nine to the X negative nine plus nine and 0 11 to the exponent zero is one. So we'll have Y equals one minus 19 which equals negative 18 or the point negative nine comma negative 18. And we can also try if X is equal to negative eight, we'll have that Y equals 11 minus 19. So same logic with that, we plugged in negative eight to the exponent X. So we'll have negative eight plus nine in the exponent which is positive 1, 11 to the, to the exponent one is 11. So we'll have 11 minus 19, which is negative eight or the point negative eight, comma negative eight. So now we know how we're shifting the graph and we have two points that will deal with that graph. So let's graph it. So we'll have a general graph like this. We have the Y axis and the X axis. And now we look at the graph that we had and we know we're gonna be shifting it nine units left and 19 yards down. So the point that we had at zero comma one will be, will be moved all the way left. So to X uh negative nine and then units down, which means we're going to have a point at negative nine, comma negative 18. Now, we need to figure out where since we were approaching the X axis infinitely but never touching it. Now, uh now we need to figure out what that ascent to would be. So what point we would, we approach now but never touch. Well, since we were doing the X axis first and now we're moving 19 units down. That means at the point negative is the point that we will approach but never connect. So we know we're gonna approach negative 19 but never intersect it. No, as we, as we had mentioned with the Y intercept in our general formula after moving it nine units left and 19 units. Now we're gonna have that point at that's it, it'll be negative nine comma negative 18. And finally, we tried the, we tried the other point that we had tried with or X is negative eight. So we'll have that point at negative eight comma negative eight in that graph like the general or the parent function rap will extend infinitely upwards as well. So now we look at this graph, the graph that we have now and see what the domain and the range would be. So for the domain again, we look at the positive X uh and the po the possible X values and we see that it'll remain the same. So we're in extending our graph infinitely to the left and infinitely to the right. So graph would be from negative infinity to positive infinity would be our domain in our range. We look at our possible Y values and now the range is gonna change because instead of approaching zero but never touching it, we're now approaching negative 19. So again, it'll be an open bracket because we never touch it. And inside we have negative 19 now comma positive infinity because we extend infinity upwards. So that will be the answer for this question. We have our graph that approaches negative 19 but never touches it. A point at negative nine comma negative 18. Another point at negative eight comma negative eight and then extending infinitely upwards similar to our parent graph with a domain of negative infinity comma positive infinity and a range of negative 19 comma positive infinity. So I hope that was helpful. And until next time.

Graph each function. Give the domain and range. See Example 3. ƒ(x) = 2^(x+2) - 4

Key Concepts

Exponential Functions

Domain and Range

Graphing Functions

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