The supply function for a product is $p ( x ) = \frac { 1 } { 3 } x ^ { 2 } + 50$ where x is the number of thousands of units a manufacturer will supply if the price is p(x) dollars. Find the inverse of this function.
A) $p ^ { - 1 } ( x ) = \frac { 1 } { 3 } \sqrt { x } + 50$
B) $\mathrm { p } ^ { - 1 } ( \mathrm { x } ) = 3 \sqrt { \mathrm { x } } - 50$
C) $\mathrm { p } ^ { - 1 } ( \mathrm { x } ) = 3 \sqrt { ( \mathrm { x } - 50 ) }$
D) $p ^ { - 1 } ( x ) = \sqrt { 3 ( x - 50 ) }$

Question

The supply function for a product is $p ( x ) = \frac { 1 } { 3 } x ^ { 2 } + 50$ where  x  is the number of thousands of units a manufacturer will supply if the price is  p(x)  dollars. Find the inverse of this function.
A) $p ^ { - 1 } ( x ) = \frac { 1 } { 3 } \sqrt { x } + 50$
B) $\mathrm { p } ^ { - 1 } ( \mathrm { x } ) = 3 \sqrt { \mathrm { x } } - 50$
C) $\mathrm { p } ^ { - 1 } ( \mathrm { x } ) = 3 \sqrt { ( \mathrm { x } - 50 ) }$
D) $p ^ { - 1 } ( x ) = \sqrt { 3 ( x - 50 ) }$

Quizplus · Accepted Answer

To find the inverse function, we first replace $p(x)$ with $y$, giving us $y = \frac{1}{3}x^2 + 50$. Then, we solve for $x$ in terms of $y$. Swapping $x$ and $y$ gives us $x = \frac{1}{3}y^2 + 50$, but we need to solve for $y$, so we rearrange to get $y^2 = 3(x - 50)$. Taking the square root of both sides gives $y = \sqrt{3(x - 50)}$, which corresponds to the inverse function $p^{-1}(x) = \sqrt{3(x - 50)}$.

The Supply Function for a Product Is p(x)=13x2+50p ( x ) = \frac { 1 } { 3 } x ^ { 2 } + 50p(x)=31​x2+50

The Supply Function for a Product Is $p ( x ) = \frac { 1 } { 3 } x ^ { 2 } + 50$