Math Is Fun Forum

Hannibal lecter

Member

Registered: 2016-02-11

Posts: 392

composed function

if 

and 

the book solution :-

why is that? it should be

  which is 

because

  is wherver I find x I should double it so when I found

I should replace the x by x^2
hence will be

am I right? If I'm not right why how should I look to the problem.

Last edited by Hannibal lecter (2022-02-01 10:26:03)

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Wisdom is a tree which grows in the heart and fruits on the tongue

Bob

Administrator

Registered: 2010-06-20

Posts: 10,623

Re: composed function

hi

Here's the way I think of functions.  A function is a number machine that has one** input and one output.  You put a number through the machine and a number pops out. 

f is the function that squares the number you put through it, and g is the function that adds 1.

Here's a diagram that shows what happens when you first apply f, and then follow that with g on the result.

Bob

** You can also have functions with more than one input but let's worry about those later.

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Hannibal lecter

Member

Registered: 2016-02-11

Posts: 392

Re: composed function

thanks, and how to think in reverse? for example :

the answer of 36. is f(x) = x^3   , g(x) = x + 1
and Answer of 39. is f(x) = e^x  , g(x) = 2x

can you tell me what is the step of thinking to extract the inside function and outside function?
why not g(x) = x^3 , and f(x) = x+1 is that possibles. it's very difficult to determine them in just looking at them please tell me how to think to find them in any question

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Wisdom is a tree which grows in the heart and fruits on the tongue

Bob

Administrator

Registered: 2010-06-20

Posts: 10,623

Re: composed function

Ok.

h(x) = (x+1)^3

The rules of algebra are just the same as the rules of arithmetic.  So, if you ever have trouble with some algebra, try it with numbers first.

Let's say x = 7 and you are asked to evaluate h(7).

You'd do 7 + 1 = 8 and then 8^3 = 512.

You wouldn't do 7^3 = 343 and then 343 + 1

So the 'thing' that happens first is the "add 1" function.  f(g(x)) is asking for g first so g(x) = x + 1, and then f(x) = x^3

If you get these the wrong way round you'd have x^3 + 1

h(x) = e^(2x)

Note that the 2x is the index but to make that clear when I use ^ for 'raise to the power' I need brackets to enclose the 2x

This tell me that doubling has to happen before raising e to the power of.  So g(x) = 2x and f(x) = e^x

If these are reversed you get 2.e^x

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob