How to shift the function xe^x mathematically by one unit on the x axis?

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To shift the function xe^(-x) one unit to the right on the x-axis, the transformation involves replacing x with (x-1), resulting in the new function (x-1)e^(-(x-1)). This method applies universally, as demonstrated with the simpler case of x^2, which becomes (x-1)^2 when shifted. Mathematically, if a function f(x) has a root at x=a, then the shifted function f(x-h) will have a root at x=a+h, confirming the shift. This illustrates the general principle of horizontal shifts in functions. Understanding this transformation is essential for manipulating functions in calculus and algebra.
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Hi Folks,

I have a function x {e^{-x}} and if i shift it one unit to the right on the x-axis we have (x-1) {e^{-(x-1)}}

How do I show this mathematically?

Even consider the simple case of x^2, if we shift by 1 unit to the right it becomes (x-1)^2

What is the method mathematically?

Regards
 
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Suppose $f(x)$ has a root at $x=a$, i.e., $f(a)=0$. Then $f(x-h)$ will have a root at $x-h=a$, or $x=a+h$. Thus, the function $f$ has been shifted $h$ units to the right.
 
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