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 Define \(g\) by  \(g(x)=5x-4\). If  \(g(x)=f^{-1}(x)-3\) and \(f^{-1}(x)\) is the inverse of the function \(f(x)=ax+b\), find \(5a+5b\).

 Nov 1, 2019
 #1
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g(x)  =  5x  - 4

g(x)  = f-1(x) - 3

g(x) + 3  = f-1(x)

 

If f(x)  = ax + b

Then

y =ax + b

y - b  = ax

(y - b)  / a  = x

(x - b)  / a  =  y  = f-1(x)

 

So

 

g(x)  + 3  =  (x - b) / a

5x - 4 + 3  =  (1/a)x - b/a

5x - 1  =  (1/a)x - b/a

 

So

5  = 1/a

a  =1/5

And

-1  = -b/a

-1  = -b / (1/5)

1  = b / (1/5)

b  =1/5

 

So

5a + 5b  =

5(1/5) + 5(1/5)  =

1  + 1  =

2

 

 

cool cool cool

 Nov 1, 2019

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