Given the quadratic function q ( x ) = − 2 x 2 + 4 q(x) = -2x^2 + 4 q ( x ) = − 2 x 2 + 4 , which is divided into two one-to-one functions, identify the function q 2 ( x ) q_2(x) q 2 ( x ) for the right side of the parabola and find its inverse q 2 − 1 ( x ) q_2^{-1}(x) q 2 − 1 ( x ) . Also, specify the domain and range for both functions.

q 2 ( x ) = − 2 x 2 + 4 q_2(x)=-2x^2+4 q 2 ( x ) = − 2 x 2 + 4 , q 2 − 1 ( x ) = 4 − x 2 q_2^{-1}(x)=\sqrt{\frac{4-x}{2}} q 2 − 1 ( x ) = 2 4 − x <path d="M983 90 l0 -0 c4,-6.7,10,-10,18,-10 H400000v40 H1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7 s-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744 c-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30 c26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722 c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5 c53.7,-170.3,84.5,-266.8,92.5,-289.5z M1001 80h400000v40h-400000z"> , Domain of q 2 : x ≥ 0 q_2:x\ge0 q 2 : x ≥ 0 , Range of q 2 : y ≤ 4 q_2:y\le4 q 2 : y ≤ 4 , Domain of q 2 − 1 : x ≤ 4 q_2^{-1}:x\le4 q 2 − 1 : x ≤ 4 , Range of q 2 − 1 : y ≥ 0 q_2^{-1}:y\ge0 q 2 − 1 : y ≥ 0

q 2 ( x ) = − 2 x 2 + 4 q_2(x)=-2x^2+4 q 2 ( x ) = − 2 x 2 + 4 , q 2 − 1 ( x ) = − 4 − x 2 q_2^{-1}(x)=-\sqrt{\frac{4-x}{2}} q 2 − 1 ( x ) = − 2 4 − x <path d="M983 90 l0 -0 c4,-6.7,10,-10,18,-10 H400000v40 H1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7 s-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744 c-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30 c26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722 c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5 c53.7,-170.3,84.5,-266.8,92.5,-289.5z M1001 80h400000v40h-400000z"> , Domain of q 2 : x ≤ 0 q_2:x\le0 q 2 : x ≤ 0 , Range of q 2 : y ≤ 4 q_2:y\le4 q 2 : y ≤ 4 , Domain of q 2 − 1 : x ≤ 4 q_2^{-1}:x\le4 q 2 − 1 : x ≤ 4 , Range of q 2 − 1 : y ≤ 0 q_2^{-1}:y\le0 q 2 − 1 : y ≤ 0

q 2 ( x ) = − 2 x 2 + 4 q_2(x)=-2x^2+4 q 2 ( x ) = − 2 x 2 + 4 , q 2 − 1 ( x ) = x − 4 2 q_2^{-1}(x)=\sqrt{\frac{x-4}{2}} q 2 − 1 ( x ) = 2 x − 4 <path d="M983 90 l0 -0 c4,-6.7,10,-10,18,-10 H400000v40 H1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7 s-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744 c-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30 c26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722 c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5 c53.7,-170.3,84.5,-266.8,92.5,-289.5z M1001 80h400000v40h-400000z"> , Domain of q 2 : x ≥ 0 q_2:x\ge0 q 2 : x ≥ 0 , Range of q 2 : y ≥ 4 q_2:y\ge4 q 2 : y ≥ 4 , Domain of q 2 − 1 : x ≥ 4 q_2^{-1}:x\ge4 q 2 − 1 : x ≥ 4 , Range of q 2 − 1 : y ≥ 0 q_2^{-1}:y\ge0 q 2 − 1 : y ≥ 0

q 2 ( x ) = − 2 x 2 + 4 q_2(x)=-2x^2+4 q 2 ( x ) = − 2 x 2 + 4 , q 2 − 1 ( x ) = − x − 4 2 q_2^{-1}(x)=-\sqrt{\frac{x-4}{2}} q 2 − 1 ( x ) = − 2 x − 4 <path d="M983 90 l0 -0 c4,-6.7,10,-10,18,-10 H400000v40 H1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7 s-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744 c-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30 c26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722 c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5 c53.7,-170.3,84.5,-266.8,92.5,-289.5z M1001 80h400000v40h-400000z"> , Domain of q 2 : x ≥ 0 q_2:x\ge0 q 2 : x ≥ 0 , Range of q 2 : y ≥ 4 q_2:y\ge4 q 2 : y ≥ 4 , Domain of q 2 − 1 : x ≥ 4 q_2^{-1}:x\ge4 q 2 − 1 : x ≥ 4 , Range of q 2 − 1 : y ≤ 0 q_2^{-1}:y\le0 q 2 − 1 : y ≤ 0

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6PRACTICE PROBLEM

Given the quadratic function $q(x) = -2x^2 + 4$ , which is divided into two one-to-one functions, identify the function $q_2(x)$ for the right side of the parabola and find its inverse $q_2^{-1}(x)$ . Also, specify the domain and range for both functions.

ANSWERS OPTIONS

$q_2(x)=-2x^2+4$ , $q_2^{-1}(x)=\sqrt{\frac{4-x}{2}}$ , Domain of $q_2:x\ge0$ , Range of $q_2:y\le4$ , Domain of $q_2^{-1}:x\le4$ , Range of $q_2^{-1}:y\ge0$

$q_2(x)=-2x^2+4$ , $q_2^{-1}(x)=-\sqrt{\frac{4-x}{2}}$ , Domain of $q_2:x\le0$ , Range of $q_2:y\le4$ , Domain of $q_2^{-1}:x\le4$ , Range of $q_2^{-1}:y\le0$

$q_2(x)=-2x^2+4$ , $q_2^{-1}(x)=\sqrt{\frac{x-4}{2}}$ , Domain of $q_2:x\ge0$ , Range of $q_2:y\ge4$ , Domain of $q_2^{-1}:x\ge4$ , Range of $q_2^{-1}:y\ge0$

$q_2(x)=-2x^2+4$ , $q_2^{-1}(x)=-\sqrt{\frac{x-4}{2}}$ , Domain of $q_2:x\ge0$ , Range of $q_2:y\ge4$ , Domain of $q_2^{-1}:x\ge4$ , Range of $q_2^{-1}:y\le0$

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