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 h(x) = e^x-1+2

Graph of the exponential function f(x) = e^x with key points and horizontal asymptote y = 0 shown.

Accepted Answer

Hello, everyone in this video, we're going to be looking at this practice problem where we want to grab G of X is equal to E to the X minus nine plus three using the reference graph of F of X which is equal to E to the X. And once we graphic, we want to determine the ascent towards equations and find the domain and range. So in this case, because we're going to be building G of X off of F of X, let's take a look at what's different between the two functions. So they both have, they both are exponential functions. You can see that the input or the X is in the exponent in both of the functions. The only difference is that G of X has a -9 in the exponent And then a plus three after the exponents. So when we think of building G of X off of F of X, we want to think about what the negative nine and the plus three are doing to the function. So in this case, the -9 is in the exponent region of the function. And in this case, it's affecting V X. So I know that it's gonna be affecting the horizontal scale or the horizontal area of G of X, I'll just say horizontal or X values and Because it's -9 and it's in the exponent instead of being the classic negative numbers to the left and positive to the right, because it's in the exponent, I have to think opposite. And when it's negative, that means I have to go to the right nine units. So whenever there's a plus or minus in the exponent of an exponential think opposite and now going a little bit different. Let's think about the plus three. Well, the plus three is not affecting the X anymore. It's affecting that vertical region or the Y values. And in this case, because it's not in the exponent, you can think of the plus three as a classical, the classical method where positive means up and negative means down. In this case, we have to go up three units. So based off of the differences that we determined What happens to G of X compared to F of X is only moving the function right nine units and then up. So if I use this first graph to help me visualize what I expect G of X to look like based off of affects, I would go right nine units. So I'm going to say this point and go right nine units. So and then Up three units. So 123. So that would be this point here. And if I want to move The starting line, I would go up again, three units, 123. And I would expect G of X to look something like this. So I went up three units and to the right nine units. And just based off of that, you can see that answer choice A does not have the expected shape or the expected transformation for GLX. So answer choice A would not be our answer. But we're looking for this sort of blue, blue line to be our G of X. And if we look at B B has the shape that we would expect. And actually, if I go back to A, you can see that I didn't move up three units, I actually moved up to 46 units. So if I moved up three units, it would be 23 solution would be here and then up. So my skill was a little bit off, But now three units up And nine units to be right. And B has that scale that we're expecting or the transformations we're expecting where it goes up three units and 24689, nine units to the right. So answer choice B seems to have the correct shape and let's just take a look at the other answer choices to make sure. So C does go up three units but then it goes to the left nine units. So not the transformation we were expecting because we need to go to the right nine units. So right now answer choice B is looking like the correct shape. So now we can figure out the horizontal ascent toads and the domain and range. So based off of this graph, you can see that there's a sort of line here Where the y values don't go below three. And we can confirm that with the equation because if I plug in a really big negative X, for example, X equals negative 1000, I have e to the negative 1000 minus nine plus three. So this becomes E to the negative 1009 plus three. And I can rewrite this negative exponent as a fraction. So it'd be one over E to the negative to the positive. Now because it's in fraction form E to the plus three. So in this case, you can see that although this fraction as my negative values get more negative, this fraction will only get closer to zero. But because I have the plus three, I can't go below three for Y values. So that means that I have a horizontal ascent tote at X equals three. And we can see that both in the equation and in the graph. So now looking at our domain, that's the range of X values that ensure that we got a defined function. And we already saw that we can plug in really big negative numbers into X. And we get a defined function, it just turns into a fraction. And similarly, we can plug in really high values for X and it just makes the exponential exponential function grow bigger. So that means that for our domain X can actually be negative infinity to positive infinity or all values. Now looking at our range, that's the possible outputs. And we've already discussed the limitation that because of the plus three, the Y values can't ever go below three. And we can see that on the graph because there's no Y values for G F X in the region below Y equals three. So that means that the range and do parentheses because Y can get really close to three but not equal three exactly because one divided by each of the 1009, although it gets really close to zero, it will never be zero. So that means that y can get really close to three but never be three, but it can grow really big so it can go to positive infinity. So these become my final answers for the horizontal ascent totes and the domain and the range. And as we discussed, answer choice B has the shape that we're expecting for this graph. And it also has the domain listed as negative infinity to positive infinity, the range as three to positive infinity and the horizontal cento at Y equals three. So hopefully this video helped and see you next time.

Key Concepts

Exponential Function and Its Graph

Transformations of Functions

Asymptotes, Domain, and Range of Exponential Functions

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