In Exercises 25-34, begin by graphing f(x) = 2^x. Then use transf...

Question

In Exercises 25-34, begin by graphing f(x) = 2^x. Then use transformations of this graph to graph the given function. Be sure to graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn graphs. g(x) = 2.2^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 seven times seven to the X using the graph of F of X Which is equal to seven x. And we want to use F of X to guide our graph for G of X. And once we have that, we want to graph the ascent oats and provide the equations for those ascent oats and then figure out the domain and range of the functions. So in this case, in order to graph F of X is equal to seven X, we can start by doing input output. So say X, we plug in certain X values and Y values. So I plug in zero into F of X, we get seven to the zero which is equal to one. If I plug in one, I get seven to the one which is equal to seven, I plug in two, I get seven to the two which is equal to seven times seven which is 49. So I have these sort of key points. Now, I also want to try -1. If I plug in 7 to the -1, I get 1/7. So you can see that once I go into the negative numbers, I start to get these fractions. So let's try negative 27 to the negative two is the same as 1/7 squared, which is equal to 1/49. So if I want to graph F of X say I draw Y access and then an X axis here. And I'm just gonna call these intervals of seven, I'll say this is 7 14, 35. Then same going down negative seven, negative 14, negative 28 -35. And then these will be normal units. So 123 and negative one, negative two, negative three. So if I grab half of X at zero, half of X is equal to one. So somewhere around here at one F of X is equal to seven At two f of X is equal to 49. So at some point over here that is outside of my craft's range, negative one X is equal to 1/ and negative two F of X is equal to one 49th. So even smaller, that's where you can see the sort of exponential function take place where it really starts to grow once we get into those positive values. So I would expect the graph of F of X to look something like that. And when we think about G of X. The only difference between F of X and G of X is that G of X has this seven multiplier in front of The exponential, which is 7 to the X. So that means that all the values of affects get multiplied by seven. You can also think of G of X as seven times F of X. So I can add a sort of column here where I have G of X, I have X and Y values. And I don't need to pull that much from math because I can just multiply these Y values from F of X by seven to get the values for G of X. So at zero F of X is equal to one. If I multiply that by seven, That means that G of X is equal to seven at At one f of X is equal to seven. If I multiply that by seven to get G of X, I get seven times seven, which is 49 I'll skip down to negative one negative one F of X is equal to 1/7. If I multiply that by seven, I get one. So that means that G of X is at one X equals -1. So now if I go back to my graph and start to build a picture for G of X, I can say at zero G of X is equal to seven at one G of X is equal to 49 a negative one, G of X is equal to one. And so I can form the sort of shape that G of X would have. And you can see that G of X is to the left of F of X. So that's the shape I'm looking for in the answer choices. And you can see that A already has a different sort of shape. So I know that A is probably not the correct answer. You look at B once again, we see a weird shape where G of X is going down and that's not the transformation we are expecting. So B is also incorrect. And if we look at sea, you can see that G of X is to the left of F of X as we expected. And we can do a little check where we know that at X equals zero G of X should be at seven. And you can see that the function or the line does pass through there. So now using this graph, we can determine the horizontal ascent toads and the domain and range. So in this case, you can see that the graph doesn't have any Y values in the negative region. And we can also confirm that using the equation because if we plug in negative values into X, say negative 1000, this becomes seven times seven to the negative 1000, which is the same as seven times 1/7 to the 1000 power and you can see that this fraction one divided by a really big number will get really close to zero but not equal zero. Exactly. So that means that both of these equations will get really close to zero but never equal zero. Exactly. And that's why we see this gap where there's no values in the negative region. So that means that we have a horizontal ascent At Y. Equals zero. And for the domain we want to think about all the possible values that X can take on. Well, in this case, X is in the exponent of the function. So that means that we can really plug in any value into X and the function will always be defined. So that means the X is all possible values. And we can write that in interval notation as negative infinity to positive infinity. And when we think of range that is the Y values that the function can take on or the outputs that the function can take on and looking at the graph, you can see that there's no negative values for why as we discussed previously and we already saw it in the function as well that why Can only get really close to zero and not equal zero. So that means that zero is our lower boundary. And since it can't equal zero exactly do a parentheses zero. And then from there why grows exponentially with exponential functions? So that means that why can go to positive infinity. There's no upper limit for the outputs that this equation can put out or this function can put out. So this becomes my final answer for this problem. And you can see the answer choice C has the expected shape or expected transformation. And the domain listed as negative infinity to positive infinity, the range as zero to positive infinity and the horizontal A cento at Y equals zero. So C becomes our final answer. Hopefully, this video helped and see you next time.

Key Concepts

Exponential Functions

Transformations of Functions

Asymptotes

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