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Suppose functions g and f have the properties that g(x)=3f^-1(x) and f(x)=24/(x+3). For what value of x does g(x)=15?

 Feb 11, 2019
 #1
avatar+101161 
+1

y = 24 / [ x + 3 ]

 

y [ x + 3 ] = 24

 

yx + 3y = 24

 

yx = 24 - 3y

 

x =   [ 24 -  3y ] /y       "swap" x and y

 

y =  [ 24 - 3x ] / x  = f-1(x)

 

So 

 g(x) =  3f-1(x) =    [ 72 - 9x] / x

 

So

 

g (x) = [72 -  9x ] / x = 15

 

72 - 9x = 15x

 

72 = 24x

 

x = 72/24   =  3

 

 

cool cool cool

 Feb 11, 2019
 #2
avatar+22290 
+6

Suppose functions g and f have the properties that

\(g(x)=3f^{-1}(x) \)

and

\(f(x)=\dfrac{24}{x+3}\).
For what value of x does g(x)=15?

 

\(\begin{array}{|rclcrcl|} \hline f\Big(f^-1(x)\Big) &=& x & | & \quad f^-1(x) &=& \dfrac{g(x)}{3} \\ && & | & \quad &=& \dfrac{15}{3} \\ && & | & \quad &=& 5 \\ f(5) &=& x & | & \quad f(5) &=& \dfrac{24}{5+3} \\ \dfrac{24}{5+3} &=& x \\ \dfrac{24}{8} &=& x \\ \mathbf{x} & \mathbf{=} & \mathbf{3} \\ \hline \end{array}\)

 

laugh

 Feb 11, 2019
edited by heureka  Feb 11, 2019

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