SOLUTION: Two functions f and g are such that f(x) = 1 + 2x and g(f(x)) = x + 3. What is the value of g(5)?

Algebra ->  Functions -> SOLUTION: Two functions f and g are such that f(x) = 1 + 2x and g(f(x)) = x + 3. What is the value of g(5)?      Log On


   

Question 1037620: Two functions f and g are such that f(x) = 1 + 2x and g(f(x)) = x + 3. What is the value of g(5)?
Found 2 solutions by stanbon, josgarithmetic:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
Two functions f and g are such that f(x) = 1 + 2x and g(f(x)) = x + 3. What is the value of g(5)?
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g(f(x)) = x+3
g(1+2x) = 2x+4 = (1+2x)+3
If 1+2x = 5, then g(5) = 5+3 = 8
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Cheers,
Stan H.
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Answer by josgarithmetic(39618) About Me  (Show Source):
You can put this solution on YOUR website!
Both functions are linear in x.

Expect g(x) to be a linear function in x, and g might be kx+p, so start it as g%28x%29=kx%2Bp.

g%28f%28x%29%29=k%28f%28x%29%29%2Bp
g%28f%28x%29%29=k%281%2B2x%29%2Bp
g%28f%28x%29%29=k%2B2kx%2Bp
g%28f%28x%29%29=2kx%2Bk%2Bp

Given is the composition g%28f%28x%29%29=x%2B3.

The two composition functions must be equal.
2kx%2Bk%2Bp=x%2B3


Identify and equate the corresponding parts.
system%282k=1%2Ck%2Bp=3%29
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system%28k=1%2F2%2Ck%2Bp=3%29

p=3-k
p=3-1%2F2
p=2%261%2F2


Return to the more variablized function g%28x%29=kx%2Bp, and substitute the values found.
highlight%28g%28x%29=x%2F2%2B2%261%2F2%29