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Jlenok [28]
3 years ago
8

Y = 2f(g(x))

Mathematics
1 answer:
IrinaK [193]3 years ago
7 0
Yes you used the chain rule properly to follow the correct steps to get the right answer. Great job.

If you wanted, you can come up with examples for f(x) and g(x) to help confirm the answer. A quick way to do this is to use something like GeoGebra to help graph the two expressions and you'll notice that the curves match up perfectly (indicating equivalent expressions). Note: GeoGebra can handle derivatives through the Derivative[] comand or you can type the function in the input bar with a tickmark after it to tell GeoGebra to derive the function.
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Consider f: N → N defined by f(0)=0 and f(n)=n-1 for all n>0.

Step-by-step explanation:

First we will prove that f is surjective. Let y∈N be any natural number. Define x as the number x=y+1. Then x∈N, and f(x)=x-1=(y+1)-1=y.  We conclude that f is surjective.

However, f is not injective. Take x1=0 and x2=1. Then x1≠x2 but f(x1)=0 and f(x2)=x2-1=1-1=0. We have shown that there are two natural numbers x1,x2 such that x1≠x2 but f(x1)=f(x2), that is, f is not injective.

Note:

If 0∉N in your definition of natural numbers, the same reasoning works with the function f: N → N defined by f(1)=1 and f(n)=n-1 for all n>1. The only difference is that you consider x1=1, x2=2 for the injectivity.

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sergey [27]
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