Inverse

Algebra Level pending

Let f ( x ) = − p x − 3 − q x + 3 f(x) = \frac{-px-3}{-qx+3} and let g ( x ) g(x) be the inverse of f ( x ) . f(x). If the point ( 7 , − 22 ) (7, -22) is on both of the graphs y = f ( x ) y = f(x) and y = g ( x ) , y = g(x), then what is the value of p + q ? p + q?

39 11 \frac{39}{11} 36 11 \frac{36}{11} 30 11 \frac{30}{11} 33 11 \frac{33}{11}

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Mahindra Jain by Mahindra Jain

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1 solution

Tom Engelsman
Nov 15, 2020

The inverse function g ( x ) = f − 1 ( x ) g(x) = f^{-1}(x) computes to g ( x ) = 3 x + 3 q x − p g(x) = \frac{3x+3}{qx-p} . If f ( 7 ) = g ( 7 ) = − 22 f(7) = g(7) = -22 , then we have:

− 7 p − 3 − 7 q + 3 = − 22 ; \frac{-7p-3}{-7q+ 3} = -22;

3 ( 7 ) + 3 7 q − p = − 22 \frac{3(7)+3}{7q-p} = -22 ;

or p = 3 , q = 3 11 ⇒ p + q = 36 11 . p = 3, q = \frac{3}{11} \Rightarrow p+q = \boxed{\frac{36}{11}}.

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