How can i prove that $h (x) = f (x - 5)$ $h(x)=f(x-5)$ when i know that $f (x - 3) = g (3 x - 2)$ $f(x-3)=g(3x-2)$ , $g (3 x + 1) = h (x + 3)$ $g(3x+1)=h(x+3)$ ??

Question

1315 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-05-31T09:46:39

See proof below

In the first equation

$f (x - 3) = g (3 x - 2)$ $f(x-3)=g(3x-2)$

Replace $x$ $x$ by $(x + 3)$ $(x+3)$

$f (x + 3 - 3) = g (3 (x + 3) - 2)$ $f(x+3-3)=g(3(x+3)-2)$

$f (x) = g (3 x + 9 - 2) = g (3 x + 7)$ $f(x)=g(3x+9-2)=g(3x+7)$

$f (x) = g (3 x + 7)$ $f(x)=g(3x+7)$ ........... $(1)$ $(1)$

In the second equation,

$g (3 x + 1) = h (x + 3)$ $g(3x+1)=h(x+3)$

Replace $x$ $x$ by $(x + 2)$ $(x+2)$

$g (3 (x + 2) + 1) = h (x + 2 + 3)$ $g(3(x+2)+1)=h(x+2+3)$

$g (3 x + 6 + 1) = h (x + 5)$ $g(3x+6+1)=h(x+5)$

$g (3 x + 7) = h (x + 5)$ $g(3x+7)=h(x+5)$ ........... $(2)$ $(2)$

Combining equations $(1)$ $(1)$ and $(2)$ $(2)$ , we get

$h (x + 5) = f (x)$ $h(x+5)=f(x)$

Replace $x$ $x$ by $(x - 5)$ $(x-5)$

$h (x - 5 + 5) = f (x - 5)$ $h(x-5+5)=f(x-5)$

$h (x) = f (x - 5)$ $h(x)=f(x-5)$

$Q E D$ $QED$

How can i prove that h(x)=f(x−5)h(x)=f(x-5) when i know that f(x−3)=g(3x−2)f(x-3)=g(3x-2) , g(3x+1)=h(x+3)g(3x+1)=h(x+3) ??