SOLUTION: If f(g(x)) = 16x+7 and g(x)= 2x-1, find f(x)

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Question 1199433: If f(g(x)) = 16x+7 and g(x)= 2x-1, find f(x)

Found 2 solutions by math_tutor2020, ikleyn:
Answer by math_tutor2020(3816) (Show Source):
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Assume f(x) is linear since f(g(x)) and g(x) are linear.
f(x) = mx+b
m = slope
b = y intercept

Now apply function composition.
f(x) = mx+b
f( g(x) ) = m*( g(x) )+b
f( g(x) ) = m*( 2x-1 )+b
f( g(x) ) = 2mx - m + b
f( g(x) ) = (2m)x + (- m + b)

The last equation has slope 2m.
The given equation f(g(x)) = 16x+7 has slope 16.

Equate the slopes to solve for variable m.
2m = 16
m = 16/2
m = 8

The equation we found for f(g(x)) has y intercept of -m+b
The given equation f(g(x)) has y intercept 7

Equate the items, apply substitution, and solve for b.
-m+b = 7
-8+b = 7
b = 7+8
b = 15

Therefore,
f(x) = mx+b
f(x) = 8x+15

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Check:

f(x) = 8x+15
f( g(x) ) = 8( g(x) )+15
f( g(x) ) = 8( 2x-1 )+15
f( g(x) ) = 8(2x)+8(-1)+15
f( g(x) ) = 16x-8+15
f( g(x) ) = 16x+7
This verifies the answer.

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Answer: f(x) = 8x+15

Answer by ikleyn(52747) (Show Source):
You can put this solution on YOUR website!