Question 1199433
<font color=black size=3>
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


-----------------------------------------


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.



-----------------------------------------


Answer: <font color=red>f(x) = 8x+15</font>
</font>