The figure shows the graph of f(x) = e^x. In Exercises 35-46, use...

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 gonna be looking at this practice problem where we want to graph G of X is equal to 13 times E to the X using transformations and then give the ascent towards equations and the domain and range of the function. And we're given the graph of F of X which is equal to E to the X. So in this case, because answer choice A is right here, I'm just going to use this graph to sort of give or put my understanding of what the equation G of X should look like. So in this case, you can see that we have F of X and using transformations, you can see that one of the only differences between the equation were given for F of X which is equal to E to the X And G of X is that G of X has this 13 in front of the E of X. So when we think about transformations and this coefficient in front of the exponential value each of the X, because 13 is greater than one, the graph or G of X will be vertically stretched by factor of compared to half of X. So in this case, if we look at our Kovacs equation, You can see that it seems to pass through the .01. So that means that our equation G of X should pass through the point 01, but stretched by a factor of 13, we have to multiply one times Go back 13. So we would pass through A line up here since 13 is not visible, it would pass through zero 13. In this case, we can see that answer choice A is already showing the wrong shape since we don't have a reflection because a reflection would have some sort of negative. So we would have the negative in this case because it's reflected across the Y axis, we would have a negative in this input. But we don't have that we don't have any new negatives. So there should be no reflections, which means we expect the same shape. So if I were to draw G of X, it should have some sort of shape similar to half of X but just steeper. So if we look through the answer choices, you can see that B once again has some sort of reflection this time across the X axis. So B is not one of the options we look at C well, we see the correct shape that we want. So now we can continue working off of this one. And the other part of the equation was figuring out the horizontal ascent totes. So if we look at this graph, you can see that these exponential equations seem to not go negative. And this makes sense because when you think about e to the negative one, this is really just a fraction that is saying one over E, so we can't get any values that are negative in exponential functions. And similarly, we can't get values that are equal to zero. Because you can see that even if I do E to the negative 1000, this is still one divided by E To the 1000. So we just get really, really close to zero, but we don't get zero, which means that there is a horizontal ascent. So you can see here At Y. Equals zero. So why equal zero is the equation for our horizontal ascent and looking at our graph, we can also determine our domain. So our domain is the range of X values that give a define answer. And in this case for E to the X or 13 times each of the X, we can plug in any value into X and we'd get a defined function. So that means that the domain would be negative infinity to positive infinity. And when we think about the range that is what are the possible outputs that we can get. And as a, as we discussed earlier because we have an exponential function, we can't get negative values or even zero because no matter how big of an X we plug in, we are just going to keep getting a smaller number. So that means that for our range, we can go from zero and notice how I have a parentheses on the zero because we can't equal zero. Exactly. We can just get really close to it all the way to positive infinity. Because with the exponential function, it can get really small, but it could also get really big. So we can go all the way to positive infinity. So this becomes the final answer here where you see that C has the shape that we expect with G of X having a steeper climb up because it needs to hit the 13, which is not really listed here, but you can draw it in and we can see that it's steeper than F of X. And the horizontal A cento at Y equals zero as well as the Domain and the range where the domain is negative infinity to positive infinity and the range is 0 to positive infinity. So C becomes our final answer. And hopefully this video helped see you next time.

Key Concepts

Exponential Functions

Transformations of Functions

Asymptotes

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