December 16th, 2016, 09:21 AM  #1 
Newbie Joined: Dec 2016 From: Pakistan Posts: 14 Thanks: 0  Multiplying f(x) with constant and x with constant is same thing?
I am confused between multiplying whole function by constant and multiplying constant with its input. If let say there is a function defined as: $$F(x) = x$$ I can also write it like this in piecewise manner: $$ f(x) = \left\{ \begin{array}{lll} x & \quad x > 0 \\ 0 & \quad x = 0 \\ x & \quad x < 0 \end{array} \right. $$ Now if I multiply whole function with 0.5, I will multiply it with outputs: $$ 0.5f(x) = \left\{ \begin{array}{lll} 0.5x & \quad x > 0 \\ 0 & \quad x = 0 \\ 0.5x & \quad x < 0 \end{array} \right. $$ If I multiply only input (x) with 0.5 confusion occurs. How do I write it as above? $f(0.5x) = 0.5x$, this is what I can write I don't know how it is going to be in piecewise function, will it be same as multiplying whole function by constant? And their graph are going to be same right? Thanks 
December 16th, 2016, 09:27 AM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 1,976 Thanks: 1026 
You picked a bad example to demonstrate that multiplying a function by a constant and multiplying its input by a constant are two different things. Consider instead $f(x)=x^2$ $f(a x) = (a x)^2 = a^2 x^2 = a^2 f(x) \neq a f(x)$ In general $a f(x) \neq f(a x)$ There is a class of functions for which $a f(x) = f(a x)$ can you determine what it is? Last edited by skipjack; December 17th, 2016 at 04:46 AM. 
December 16th, 2016, 09:43 AM  #3  
Math Team Joined: Jan 2015 From: Alabama Posts: 3,194 Thanks: 871  Quote:
Quote:
Or possibly you just copy and pasted the mistake above. "To err is human, to really screw up requires a computer." Quote:
Last edited by skipjack; December 17th, 2016 at 04:49 AM.  
December 17th, 2016, 04:04 AM  #4  
Newbie Joined: Dec 2016 From: Pakistan Posts: 14 Thanks: 0  Quote:
Last edited by skipjack; December 17th, 2016 at 04:50 AM.  
December 17th, 2016, 04:26 AM  #5 
Newbie Joined: Dec 2016 From: Pakistan Posts: 14 Thanks: 0 
Thanks romsek and Country boy it helped! I had bad example and that rule is true generally. I thought it is always true. I have one more question about it. Let consider new function like $f(x) = x^3$. If I do these two things: $0.5f(x) = 0.5x^2$ $f(0.5x) = (0.5x)^2$ If I plot its graph, both are change but there is a common effect. Graph compresses vertically. Compression is different for both cases but it do compress. If we take another look we can also say that it stretches horizontally in both cases. 1st one does less than second though. If constant is greater than zero it will stretch vertically or in other words, it compresses horizontally. This is what I understood If you compress along one axis it will stretch along other axis. But websites tell me: "We can stretch or compress it in the xdirection by multiplying x by a constant. If constant C is > 1 it compresses or if 0<C<1 it stretches" And "We can stretch or compress it in the ydirection by multiplying the whole function by a constant. C > 1 stretches it, 0 < C < 1 compresses it" But If you plot graphs it says something much easier according to my understanding: No matter if you multiply x or whole function with constant, if its 0<C<1 it will compress along y axis or if C>1 it stretches along y axis No matter if you multiply x or whole function with constant, if its 0<C<1 it will stretch along x axis or if C>1 it compress along x axis This is what I think happens but websites say it about different axis depending on what is being multiplied i.e. whole function or x. Am I missing something? I am posting graph in next post. 
December 17th, 2016, 04:31 AM  #6 
Newbie Joined: Dec 2016 From: Pakistan Posts: 14 Thanks: 0 
Graph for 0<C<1, it compresses along y axis: Graph for C>1, it stretches along y axis: Both made from Desmos Graphing Calculator 
December 17th, 2016, 05:04 AM  #7 
Global Moderator Joined: Dec 2006 Posts: 19,057 Thanks: 1618 
8x³ and (2x)³ are the same, but what if you used a function such as x²  x³?

December 17th, 2016, 05:19 AM  #8 
Newbie Joined: Dec 2016 From: Pakistan Posts: 14 Thanks: 0  Yes they are same..... I am getting same results as I described. If multiplied x or whole function with constant, it will have similar effect on graph with different degree of compression/stretching.

December 17th, 2016, 08:43 AM  #9 
Senior Member Joined: Sep 2015 From: USA Posts: 1,976 Thanks: 1026  
December 17th, 2016, 08:56 AM  #10 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,364 Thanks: 100  

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