Here the idea is to express the given function in terms of (2x-3). So
2x^2 -4x + 1 = 1/2 * (4x^2 -8x + 2) = 1/2 * [(4x^2 -12x + 9) + 4x - 7] = 1/2 * [(2x-3)^2 + 2(2x -3) - 1]
continue.....
Here the idea is to express the given function in terms of (2x-3). So
2x^2 -4x + 1 = 1/2 * (4x^2 -8x + 2) = 1/2 * [(4x^2 -12x + 9) + 4x - 7] = 1/2 * [(2x-3)^2 + 2(2x -3) - 1]
continue.....
“... mathematics is only the art of saying the same thing in different words” - B. Russell
I observe that no one has suggested what to me is the best algebraic approach to this (no guessing).
You know that f(2x - 3) = 2x^2 - 4x + 1, and you want to find an expression for f(u). To do that, you can just substitute
u = 2x - 3
and solve to express x in terms of u. Put this in place of x on the right side, and simplify to obtain f(u).
Another way to describe this process, if you are comfortable with the notation of function composition, is that you want [tex]f \circ g = h[/tex], where h(x) = 2x^2 - 4x + 1 and g(x) = 2x - 3, but you don't know f. You can just compose each side with [tex]g^{-1}[/tex]:
[tex](f \circ g) \circ g^{-1} = h \circ g^{-1}[/tex]
The left side simplifies to f, so the right side is the answer.
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