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+3) +1

Accepted Answer

Hello everybody. I have everything 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 where our function is F is equal to 11 to the exponent X plus three. And then outside of the exponent plus 27. Now when looking at a question like this, I want to look at the function that they provided us and see if there's any way that this could focus on the parents' function. So in this case, the parent function would be 11 to the exponent X. So this is the function that has no transformations to it. So by just having 11 to the exponent X without the plus three and without the plus outside this would be the parent function, the plus three and the plus 27 function as transformations to this graph. So what would this graph look like? Well, we draw a Y axis and an X axis and we know uh some points for this graph. So for example, if we have an X of zero, we have a Y of one. That is the why exponent the, the Y intercept. And this is because if you plug in zero for the X, you have 11 to the exponent zero, which is one. And now we also, we can also plug in the export the X of one and I would have a Y of because once again, if you plug in one for the exponent X, you would have 11 to the first power, which is 11. So now we have two points, we have one comma 11 and the Y intercept that zero comma one. And then we can do the same thing or negative one. And if you plug in negative one for the exponent, you would have a Y of one divided by 11. And we know that the, this graph will go infinitely to the left approaching the X axis but never hitting the X axis. So the X axis would function as an as, as code in this case and then it would go infinitely upwards. After the Y axis, the Y intercept functions kind of where it's, it curves up and then goes infinitely upwards. So now we look at the domain and the range. So the domain has to do with the possible X values of this graph. In this case, like I mentioned, we're moving infinitely to the left and infinitely to the right. Although to the left, we're moving at a much faster rate than to the right. We are still moving infinitely to the right we're just moving up at a lot quicker rate. So we move from negative infinity along the X axis to positive infinity. Because like I mentioned, this graph is going infinitely to the left and to the right now, we're looking at the range, we have to look at the possible Y values. So here we mention an asil in this case, the X axis that is the same as saying zero because we're talking about Y values here, the range has to do with possible Y values and the X axis is a Y of zero. And when writing this, since we never touch the X axis, we have to write an open bracket. And then a zero, if we were to touch the X axis, it would be a closed bracket. But since we don't, we have an open bracket and then zero comma and then as I mentioned earlier, we are opening infinitely upwards. So from zero comma to positive infinity. And since we never touch infinity, there will also be an open bracket and that will be the domain arrange for the parent function. Now, using this information, we can see the transformations that we do to this graph. So for example, all right, the first transformation will be 11 X plus three and we're adding three to the exponent. Now this means that we are shifting the graph of X 11 to the exponent X three units left. And then if we add the other transformation. So we'll have 11 X plus three, X plus three being in the exponent. And outside of all that, we have a plus 27 this means that we are shifting graph since the 27 is outside, we're shifting the graph of 11 to the exponent X plus three 27 units. Oh So using those transformations, we can reflect and draw another graph. So once again, we'll draw a Y axis and an X axis. So the first thing we can move is the Asy Tode, we see that with the adding, adding three to the exponent shifts it to the left. Which means there's no, it does, that does not affect the asym tode since we have a horizontal asci tode. However, the plus 27 outside will affect it since we're moving the graph units up, that means that we had an Asymptote first at the X axis or a Y of zero. Now, if we move it 27 units upwards, it means we're going to have a, an, the Asymptote now at a Y of 27. So, well, the graph will always be approaching 27 but never touching it. Now, we can try a couple of different points. So we can say I'm going to say if X equals negative three, then Y equals one plus 27. And this is because where you're plugging in negative three to the X negative three plus three is zero and 11 to the exponent zero is one. So one plus 27 which is 28 or the point X of negative three comma 28. And then I'll also say if X is equal to negative two will have a Y of plus 27. Once again, if we plug in the negative two to the X exponent, we'll have negative two plus three as the exponent which is positive one and to the exponent one is 11. So we'll have 11 plus 27 which is or the exponent negative two comma 38. So we can have those graphs here. So we'll, we can have a point at negative three, negative three comma and another point at negative two comma 38. And then the graph will once again move infinitely upwards. You can also try to figure out the why intercept by plugging in a, the, the, the appropriate value here. In this case, it would be an X of zero. But that will give you a very large value because you would have zero plus three in the exponent which would be 11 to the third power plus 27. So that would be a very large value. So here, what we have is just the graph of the parent function shifted three units to the left and 27 units upwards. So now for the domain, we once again look at the possible X values here. So the do, the possible domains does not change since we're still expanding infinitely to the left and infinitely to the right. So we would still be at negative infinity to positive infinity. The, the domain is not affected. However, now, when looking at the range, we saw that we moved the ascent tode from 0 to 27. So now instead of starting at zero, we're starting at 27. And once again, since it's an Asymptote, we never touch it. So it will be an open bracket and then 27 comma and we're still opening infinity upwards. So it'll be 27 comma infinity, positive infinity with another open bracket since we never touch infinity. And this would be the domain in range for our function and the graph to go along with it. 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+3) +1

Key Concepts

Exponential Functions

Domain and Range

Graphing Functions

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