Find the inverse function of g(x) = 2x2 - 1. Also, determine if 2, 3, 8, and 10 are elements of set A = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12. Lastly, solve 2^n = 4^(n-3).

Morning Art — let's get this done. Step 1: Replace g(x) with y. y = 2x^2 - 1 Step 2: Swap x and y. x = 2y^2 - 1 Step 3: Solve for y. Add 1 to both sides: x + 1 = 2y^2 Divide by 2: (x + 1)/(2) = y^2 Take the square root of both sides. Since the original function g(x) = 2x^2 - 1 is not one-to-one over its entire domain, we typically restrict the domain of g(x) to x 0 (or x 0) to find an inverse function. Assuming x 0, then y 0 for the inverse function. y = sqrt((x + 1)/(2)) Step 4: Replace y with g^-1(x). g^-1(x) = sqrt((x + 1)/(2)) The domain of g^-1(x) is x -1. The inverse function is: g^-1(x) = sqrt((x + 1)/(2)) --- a) The set A = \1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12\. The number 2 is an element of set A. 2 A b) The set A = \1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12\. The number 3 is an element of set A. 3 A c) The set A = \1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12\. The number 8 is an element of set A. 8 A d) The set A = \1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12\. The number 10 is an element of set A. 10 A --- Step 1: State the formula for the number of subsets. The number of subsets of a set with n elements is 2^n. Step 2: Set up the equation based on the given information. The problem states that the set has 4^(n-3) subsets. 2^n = 4^(n-3) Step 3: Express both sides with the same base. Since 4 = 2^2, substitute this into the equation. 2^n = (2^2)^(n-3) 2^n = 2^2(n-3) Step 4: Equate the exponents and solve for n. n = 2(n-3) n = 2n - 6 Subtract n from both sides: 0 = n - 6 Add 6 to both sides: n = 6 The number of elements in the set is n=6. Send me the next one 📸