Question 115390: please! I need help solving a couple homework question.
1. Find a formula for the inverse of f(x)= 4x-3
2. find g o f (x) if f(x)=x^2 +3x and g(x)=2x-1
3. Convert b=log37 (the 3 is small and at the bottom of log) to an exponential equation
Answer by ganesh(20)   (Show Source):
You can put this solution on YOUR website! (1) Let g(x) be the inverse of f(x) = 4x -3.
Then, (f.g)(x) = x ( this is the property of inverse functions).
That is, f(g(x)) = x
That is, 4g(x) - 3 = x (Simply substitute g(x) in the place of x in f(x)).
That is, 4g(x) = x + 3
Or, g(x) = (x+ 3)/4.
Therefore, the inverse of f(x) = g(x) = (x+3)/4.
(2) Here f(x) = x^2 + 3x and g(x) = 2x - 1.
We have to find, (g.f)(x).
(g.f)(x) = g(f(x)) = 2(x^2 + 3x) - 1 = 2x^2 + 6x -1 (Just plug in the expression for f(x) in the place of x in g(x)).
This is the required answer.
(3) Represent log7 to the base 3 as log7(3).
We have b = log7(3).
So, 3^b = 3^{log7(3)}
Or, 3^b = 7. (general rule: a ^{logx(a)} = x)
So, the required exponential equation is, 3^b = 7
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