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?
+128406 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
+26367 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?
\(\begin{array}{|rclcrcl|} \hline f\Big(f^-1(x)\Big) &=& x & | & \quad f^-1(x) &=& \dfrac{g(x)}{3} \\ && & | & \quad &=& \dfrac{15}{3} \\ && & | & \quad &=& 5 \\ f(5) &=& x & | & \quad f(5) &=& \dfrac{24}{5+3} \\ \dfrac{24}{5+3} &=& x \\ \dfrac{24}{8} &=& x \\ \mathbf{x} & \mathbf{=} & \mathbf{3} \\ \hline \end{array}\)
