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December 16th, 2016, 10:21 AM   #1
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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
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December 16th, 2016, 10:27 AM   #2
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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?
Thanks from shahbaz200

Last edited by skipjack; December 17th, 2016 at 05:46 AM.
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December 16th, 2016, 10:43 AM   #3
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Quote:
Originally Posted by shahbaz200 View Post
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:

$$
f(x) = \left\{
\begin{array}{lll}
x & \quad x > 0 \\
0 & \quad x = 0 \\
x & \quad x < 0
\end{array}
\right.
$$
I assume it was a typo but if x < 0, |x| = -x

Quote:
Now if I multiply whole function with 0.5, I will multiply it with outputs:

$$
0.5f(x) = \left\{
\begin{array}{lll}
0.5x & \quad x > 0 \\
0 & \quad x = 0 \\
0.5x & \quad x < 0
\end{array}
\right.
$$
Perhaps it was not a typo - you have the same mistake here - if x < 0, this is -0.5x.
Or possibly you just copy and pasted the mistake above. "To err is human, to really screw up requires a computer."

Quote:
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
No, in general f(0.5x) is NOT the same as 0.5f(x). More generally, f(ax) is not necessarily equal to af(x).

Last edited by skipjack; December 17th, 2016 at 05:49 AM.
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December 17th, 2016, 05:04 AM   #4
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Quote:
Originally Posted by romsek View Post
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?
Functions in which variables have power raised to 1?

Last edited by skipjack; December 17th, 2016 at 05:50 AM.
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December 17th, 2016, 05:26 AM   #5
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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 x-direction 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 y-direction 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.
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December 17th, 2016, 05:31 AM   #6
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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
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December 17th, 2016, 06:04 AM   #7
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8x³ and (2x)³ are the same, but what if you used a function such as x² - x³?
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December 17th, 2016, 06:19 AM   #8
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Quote:
Originally Posted by skipjack View Post
8x³ and (2x)³ are the same, but what if you used a function such as x² - x³?
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.
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December 17th, 2016, 09:43 AM   #9
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Quote:
Originally Posted by shahbaz200 View Post
Functions in which variables have power raised to 1?
excellent
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December 17th, 2016, 09:56 AM   #10
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Quote:
Originally Posted by shahbaz200 View Post
Functions in which variables have power raised to 1?
Quote:
Originally Posted by romsek View Post
excellent
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