The figure shows the graph of f(x) = e^x. In Exercis...

Question

The figure shows the graph of f(x) = e^x. In Exercises 35-46, use transformations of this graph to graph each function. Be sure to give equations of the asymptotes. Use the graphs to determine graphs. each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn g(x) = 2e^x

Accepted Answer

Hello everyone. In this video, we're going to be looking at this practice problem where we want to graph G of X which is equal to E to the X plus 15 using F of X, which is equal to E to the X as reference. And once we graph it, we want to find the ascent toads equations and figure out the domain and range for G of X. So in this case, you can see that there is one difference between F of X and G of X and that is that G of X has also an exponential function E to the X, but it has a plus on the outside or after the exponential function. So we want to think about what this plus 15 is doing two G of X. So if we add plus 15 to F of X, what would that look like? Well, adding 15 because it's outside of the exponential portion and not affecting the X we know that this is going to affect the output or the Y values just say why values or the vertical, vertical region of The function. And plus 15 means that we're going to go up 15 units because we have a positive 15. So that being the only difference, we know that we're going to have a vertical translation of 15 units up from F of X. So if we look at this first graph, you can see that G of X indeed is portrayed as going up 15 units. So this could be our function. But let's take a look at the other answer choices to make sure. So in B you can see that we went down 15 units, but that's not the translation we were looking for. So b is discarded as the correct answer. In C, you can see that we went to the left 15 units, but we already determined that the 15 was affecting the vertical scale. So again, we know C is not a viable option. So it looks like a is the only option that has the correct shape that we expect. So we'll continue and find the horizontal A cento and the domain range from this graph. So when we think of ascent, we think of the line where the function approaches that number but it never passes it. So if we look at our equation G of X is equal to E to the X plus 15, you can see that if I plug in a really negative number, really big negative number into this equation, let's say negative 1000, I have E to the negative 1000 plus 15. Well, I can write E to the negative 1000 as a fraction where it would become one divided by E to the 1000 plus 15. So you can see that as we go two very big negative numbers, this fraction here gets very close to zero but not exactly zero because it's still a fraction. So although this first part gets very close to zero, we have this plus 15 in all of those scenarios. So that means that no Y value goes below zero below 15 because of the plus 15. And if we look at the graph, you can see that's true because there's no Y values for G of X below Y equals 15. So that means that we have a horizontal ascent oh At the line y equals 15. And now we can think about the domain and recall the domain is the possible X values that we can plug into this function and still get a defined answer. Well, in this case, the X is in the exponent and we already saw we could plug in really big negative numbers. And if we plugged in really big positive numbers, we would just get a really large exponent. So that tells me that X can be all possible values. And if we write that in interval notation, it would be from negative infinity to positive infinity. And now to find the range, That's the possible Y values. But we already established that there is no y values below 15. So that means I have a lower boundary at 15. And that's with the parentheses because remember my function can't ever be 15 exactly because of this fraction. And from there, we can also go all the way up to positive infinity on the Y values because this is an exponential function. And if I plug in X To the 1000, our equation would be E to the 1000 plus 15. So you can see that this exponent would be really big. So why values can go all the way to positive infinity? And again, we can see that these domains and the range in the graph Because there's no values for Y below 15. And there's no other limitations In our function g of x besides that as y equals 15. So this becomes my final answer for this problem. And you can see that answer choice A Also list the domain as negative infinity to positive infinity, the range as 15 to positive infinity And the horizontal a cento at y equals 15. So answer choice A is our final answer. Hopefully this video helped and see you next time.

Key Concepts

Exponential Functions and Their Graphs

Transformations of Functions

Asymptotes, Domain, and Range

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