Combinations of functions

[ilii]

Question:
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Graph all three of f(x), g(x), and the indicated combination on the same axis. In each case, state the domain and range of the composite function.

f(x)= (2^-x) , g(x)=sin10x,


y=(2^-x)*sin10x
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THE GRAPH: http://www.wolframalpha.com/input/?i=(2^-x)sin10x

I have a few problems:

What is the most effective way to go about graphing the combination of the two functions?

If I take 2pi/10 , I get the period of pi/5 , and then an interval of pi/20. Do I get my Y coordinates by plugging in pi/20 into the combined equation? i.e. y=(2^-(pi/20))*sin10(pi/20) ? what values am I supposed to put into x to get a y value on the graph of the function y=(2^-x)*sin10x?

If this is the proper way to find the y coordinates, after how many plotted points do I stop at? do I plot 2 full periods full of points for the combined graph?

What is the easiest way of solving something like this? is there an easy way to do this mentally? I can visualize the unit circle while I do basic functions, but not combinations of functions like this..

thank you~

 

[Simon Bridge]

If I take 2pi/10 , I get the period of pi/5 , and then an interval of pi/20.

This sentence does not make sense.

You can plot out a function just by plugging and chugging - input a set of x values at regular intervals and plot the corresponding y values - but that is, as you've noticed, inefficient.

What is the most effective way to go about graphing the combination of the two functions?

Understand the relationship the function describes, and what the key points are that will help you sketch in the lines.
Usually you only need to know the locations of the peaks, troughs, inflexion points, roots, and assymtotes... the places the graph crosses the y-axis is usually good too. With practice you learn to recognize where to look.

I think this exercize is supposed to help you gain the experience so it is limited how many hints I can give you.

 

[ilii]

This sentence does not make sense.

You can plot out a function just by plugging and chugging - input a set of x values at regular intervals and plot the corresponding y values - but that is, as you've noticed, inefficient.Understand the relationship the function describes, and what the key points are that will help you sketch in the lines.
Usually you only need to know the locations of the peaks, troughs, inflexion points, roots, and assymtotes... the places the graph crosses the y-axis is usually good too. With practice you learn to recognize where to look.

I think this exercize is supposed to help you gain the experience so it is limited how many hints I can give you.

Yes, I agree to everything you said... I am very early (pre-calc) and the material taught to me has been vague on more than one occasion - I end up having to spend most of my time searching the web trying to figure things out

In this case I am not sure if 'graph' means sketch (approximate y values, a rough idea) or if it means find the precise y values along the graph a few periods in length and label them each. Sometimes the resultant graph looks huge and it just scares me I guess. Will only knowing the zeroes, asymptotes, etc allow me to find these points? or just give me enough data to work with so I can make a sketch? Also, how many points do I need to actually plot if the graph looks like y=(2^-x)*sin10x?

ty

 

[Simon Bridge]

These days, graphing a function is done by a computer.
If you are asked to graph something by hand, then you only have to get the "important bits" just right, and the rest can be just indicated ... this would be like you'd think of as a "careful sketch". Which bits count as "important" depends on the situation - so it's a judgement call.
Gaining the ability to make good judgement calls like that is part of the point of the lesson - and it is why so much is deliberately vague.

If you were to plot a bunch of y values for a series of regularly spaced x values, then you may not plug in the right sequence of x values to get all the nuances. So you have to be clever about it. You want points closer together near critical points, and you want the actual critical point to be located exactly.

The total number of points you need is the number that allows you to show all the important information about the function.
For your problem, I'd only actually calculate 3-4 points for each one - but that is because I know what to look for,
You are just starting out so you need to explore more.

For your combined function, I will suggest that you find the x value of each peak and trough - since you have already sketched $2^{-x}$ you only need a ruler to find the y-position of the peaks, and you should already know how to sketch a sine wave.

 

[ilii]

thank you, very clear

 

[HallsofIvy]

You said "composite" of two functions, then immediately wrote the product of f and g:
fg(x)= (2^{-x})sin(10x)[/itex]. Is that what you meant or did you mean the composition of the functions:
$$f(g(x))= 2^{-sin(10x)}$$
and
$$g(f(x))= sin(10(2^{-x})$$