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34 Function Problems : Graphing, Translation, Reflection, Domains, Limits, Derivatives and Tangents

For the following graph the given functions on a computer screen, how are these graphs related?

1) Y=2^x, y=e^x, y=5^x, y=20^x

2) Y=3^x, y=10^x, y=(1/3)^x, y=(1/10)^x

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Make a sketch of the function.

7) y=4^x-3
8) y=-2x^-x
9) y=3-e^x

13) Starting with the graph of y=e^x, write the equation of the graph that results from.
a) shifting two units downwards.
b) shifting two units to the right.
c) reflecting about the x axis.
d) reflecting about the y axis.
e) ) reflecting about the x axis and then about the y axis.

Find the domain of the functions:

a) f(x)=1/(1+e^x)
b) f(x) = 1/(1-e^x)
c) g(t)=sin(e^-1)
d) g(t) = SQRT(1-2^t)

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19) Suppose the graphs of f(x)=x^2 and g(x)=2^x are drawn on a coordinate grid where the unit of measurement is 1 inch. Show that at a distance of 2 feet to the right of the origin, the height of the graph of f is 48ft but the height of the graph of g is about 265mi.
20) Compare the functions f(x)=x^10 and g(x)=e^x by graphing both f and g in several viewing rectangles. When does the graph of g finally surpass the graph of f?

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Find the limits for the following

1) lim as x approaches infinity(1.001)^x
2) lim as x approaches infinity(e)^-2x
3) lim as x approaches infinity(e^3x - e^-3x)/(e^3x + e^-3x)
4) lim as x approaches (pi/2)^+ (e^tanx)
5) lim as x approaches 2^+ (e^(3/(2-x)))
6) lim as x approaches 2^- (e^(3/(2-x)))

Differentiate the functions:

1) f(x) =x^2e^2
2) y=e^ax^3
3) f(u) =e^(1/u)
4) f(t)=e^(tsin2t)
5) y=SQRT(1+2e^3x)
6) y=e^e^x
7) y= (ae^x+b)/(ce^x+d)
8) y=e^x/(1+x)
9) g(x)=SQRT(x)e^x
10) y=e^(ktanSQRT(x))
11) y=SQRT(1+xe^-2x)

Find the equations of the tangent line to the curve at the given point.

1) y=e^(2x) cos(pi)(x), (0,1)

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Mathematics, Calculus
Year 1
Inverse functions
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Please solve each problem with a detailed solution showing each step to solve the problem. Since the symbols confuse me at times please use "baby" math to show how to get from the start to the end. I understand the book in some ways, but the more I see completed the better I can think about the rest of the problems I need to do. Also, if you type it all back to me, not a scanned print of your help, please keep things separated and neat please. Also, I am a returning adult student - you may know this stuff inside and out, but I do not. Thanks!
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Inverse function problems:

For the following graph the given functions on a computer screen, how are these graphs related?

1) Y=2^x, y=e^x, y=5^x, y=20^x
The graph of each of the function is draw on the following graph.

By looking at the graph, we see that each all the functions are exponential functions.

2) Y=3^x, y=10^x, y=(1/3)^x, y=(1/10)^x

From the graph we see that 3^x and (1/3)^x are inverse to each other. Similarly 10^x and (1/10)^x are inverse to each other.

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Make a sketch of the function.

7) y=4^x-3
Solution:
First draw the graph of y = 4^x.
If you subtract 3 from the function, then the graph 4^x -3 will be shifted 3 units down.

8) y=-2x^-x
Solution:
Here the basic function is y=x^(-x).
The graph of x^(-x) is the following.

Then multiply the function by 2, the graph is stretches by 2 units.

Now, multiply 2x^(-x) by -1. Then the graph is reflected about x axis.

9) y=3-e^x
Solution:
Here ...

Solution Summary

Thirty-four function problems are solved. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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