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Let $g(x) = 4x + 1$.

Find $f(x)$, given that the equation $(g \circ f)(x) = x + 5$ is true for all values of $x$.

From my text, it looks like this would be expressed as $g[f(x)]$, but I don't see how you can determine the definition of $f(x)$. Do I need to assume a value of x and work backwards through the problem?

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up vote 10 down vote accepted

$(g\circ f)(x) = g(f(x))$.

So you know that $g(a) = 4a+1$, no matter what $a$ is. I particular, that means that $$(g\circ f)(x) = g\bigl(f(x)\bigr) = 4(f(x)) + 1.$$ Now, you want $(g\circ f)(x) = x+5$, so that means that whatever $f(x)$ is, you want $$4(f(x)) + 1 = x+ 5.$$ Can you figure out what $f(x)$ should be for this to be true?

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Yes, this simplifies to f(x) = (x+6) \ 4, which "is" true for all values of x because x is not in the denominator. –  erimar77 Oct 18 '11 at 20:40
    
@erimar77 You mean $\frac{x+4}{4}$. –  Austin Mohr Oct 18 '11 at 20:43
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@erimar77 Also, in general, there are other issues to consider other than $x$ being in the denominator. You should pay extra attention whenever $x$ is in the denominator, $x$ is in an odd radical, or $x$ is in a logarithm. –  Austin Mohr Oct 18 '11 at 20:45
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f(x)= (X+4)/4

This is how I got it:

4(f(x))+1=x+5

4(f(x))+1-1=x+5-1

4(f(x))=x+4

(4(f(x)))/4=(x+4)/4

(f(x))=(x+4)/4 So the x that would make the equation true is (x+4)/4

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