Help with the function question please?

Question

Let $f (x) = 2 x + 7 and g (x) = 3 x + c$ $f(x) = 2x+7 and g(x) = 3x+c$ . Find $c$ $c$ if $f (g (x)) = g (f (x))$ $f(g(x)) = g(f(x))$ for all values of $x$ $x$ .

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Answer 1 · 2018-02-21T05:35:43

$f (g (x)) = g (f (x))$ $f(g(x))=g(f(x))$ if $c = 14$ $c=14$

As $f (x) = 2 x + 7$ $f(x)=2x+7$ and $g (x) = 3 x + c$ $g(x)=3x+c$

hence $f (g (x)) = f (3 x + c) = 2 (3 x + c) + 7 = 6 x + 2 c + 7$ $f(g(x))=f(3x+c)=2(3x+c)+7=6x+2c+7$

and $g (f (x)) = g (2 x + 7) = 3 (2 x + 7) + c = 6 x + 21 + c$ $g(f(x))=g(2x+7)=3(2x+7)+c=6x+21+c$

It is apparent that generally $f (g (x)) \neq g (f (x))$ $f(g(x))!=g(f(x))$

However, in this case we are having another variable $c$ $c$

and as $6 x + 2 c + 7 = 6 x + 21 + c \Leftrightarrow c = 14$ $6x+2c+7=6x+21+chArrc=14$

Hence in a particular case

when $c = 14$ $c=14$ , we may have $f (g (x)) = g (f (x))$ $f(g(x))=g(f(x))$