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Given $f (x) = 4 x + 5$ $f(x) = 4x + 5$ and $g (x) = 3 x - 8$ $g(x) = 3x - 8$ how do you find g(f(x))?

1 Answer

Aug 27, 2016

$f (g (x)) = 12 x - 27$ $color(green)(f(g(x))=12x-27$

Explanation:

Given
$XXX f (x) = 4 x + 5$ $color(white)("XXX")f(x)=4x+5$
and
$XXX g (x) = 3 x - 8$ $color(white)("XXX")g(x)=3x-8$

The problem here usually occurs because $x$ $x$ appears in both function definitions. But $x$ $x$ is just an arbitrary place holder; we can replace it with any other variable.

Let's replace $x$ $x$ in the definition of $f (x)$ $f(x)$ with $w$ $w$ (another arbitrary variable). Then we have:
$XXX f (w) = 4 w + 5$ $color(white)("XXX")f(color(red)(w))=4color(red)(w)+5$

Now $w$ $color(red)(w)$ could be replaced with anything.
In particular we could replace it with $g (x)$ $color(blue)(g(x))$ .

That is
$XXX f (g (x)) = 4 g (x) + 5$ $color(white)("XXX")f(color(blue)(g(x)))=4color(blue)(g(x))+5$

But since $g (x) = 3 x - 8$ $color(blue)(g(x)) = color(blue)(3x-8)$
we have
$XXX f (g (x)) = 4 (3 x - 8) + 5$ $color(white)("XXX")f(color(blue)(g(x)))=4color(blue)((3x-8))+5$

Simplifying
$XXX f (g (x)) = (12 x - 32) + 5$ $color(white)("XXX")f(g(x))=(12x-32)+5$

$XXX f (g (x)) = 12 x - 27$ $color(white)("XXX")f(g(x))=12x-27$

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