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We have $f(x)$. If we want to make up $f(ax+b)$ :

1.We should first create $f(ax)$ and then $f(ax+b)$

2 Firstly create $f(x+b)$ and then $f(ax+b)$ ?

In the other words , I'm in doubt when we create $f(ax)$ the coefficient of $x$ only is multiplied by $x$ and make up $f(ax+b)$ or multiple by $x+b$ and make up $f(ax + ab)$ and therefore second way is wrong.  If someone provide several examples like $f(x) = x^2$ and create $f(ax+b) = (ax+b)^2$ using function transformations is helpful. 

Example : Consider $f(x) = sin(x)$ . I used Mathematica for drawing functions. We want to create $f(3x-1)$. We should draw $sin(3x)$ and then $sin(3x-\frac{1}{3})$ but I think we have to draw $sin(3x-1)$ in last step. Am I wrong ? Why? $f(x) = sin(x)$$f(3x) = sin(3x)$f(3x-1) = sin(3x-1)

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    Are you asking about transforming the *graph* of $f$? – 2017-01-15
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    Yes , I'm looking for transforming the graph of $f(x)$ – 2017-01-15
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    For $x\mapsto ax'+b$, scale first, then shift. If you shift first instead, you have to shift by $b/a$ to get the same result. – 2017-01-15
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    @amd I had the same thought as you but for example consider $f(x) = sin(x)$. For creating $f(3x-1)$ , we should first draw $sin(3x)$ and then $sin(3x-\frac{1}{3})$ and not $sin(3x-1)$ . – 2017-01-15
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    You’re right. The shift is $b/a$ with either order of application. I really shouldn’t try to do this stuff after taking decongenstants :) – 2017-01-15
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    @amd But I'm still confused about it. I need a proof for it. – 2017-01-15
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    @amd Also it seems that the answer of A. Molendijk is wrong! – 2017-01-15

1 Answers 1

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Going from $f(x)$ to $g(x) = f(ax + b)$ requires two $2$ function transformations.

  • A horizontal movement of $-b$ by substitution $x$ by $x+b$. $(g(x))$
  • A horizontal multiplication of $\frac{1}{a}$ by substition $x$ by $ax$. $(h(x))$

Example: Let $f(x) = x^2$. Then

$$g(x) = (x+b)^2$$ and $$h(x) = (ax+b)^2.$$ Question: Does the order matter, that is, can we first do the second and then the first transformation?

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    Okay , but what is the answer to the question that you proposed ? – 2017-01-15
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    When we do a horizontal multiplication of $1/a$ we multiple only $x$ by $a$ or multiple $x+b$ by $a$ ? – 2017-01-15