Find the inverse function and determine if the relation is a function. Also, find the equation for g(x) and its inverse.

Morning sthamntungwa — let's get this done. 2.1.1 To find the inverse function f^-1(x) from a set of ordered pairs, we swap the x and y coordinates for each pair. Given f(x) = \(-2; 3), (0; -1), (1; 2), (2; 0)\. f^-1(x) = \(3; -2), (-1; 0), (2; 1), (0; 2)\ The inverse function is \(3; -2), (-1; 0), (2; 1), (0; 2)\. 2.1.2 A relation is a function if each input (x-value) corresponds to exactly one output (y-value). In the given set f(x) = \(-2; 3), (0; -1), (1; 2), (2; 0)\, all the x-values (-2, 0, 1, 2) are unique. Therefore, each x-value has only one corresponding y-value. f(x) is a function. 2.2.1 The equation for g(x) is given as y = a^x. We are given that the point A(4; 8) lies on the graph of g(x). We can substitute these coordinates into the equation to find the value of a. Step 1: Substitute the coordinates of A(4; 8) into y = a^x. 8 = a^4 Step 2: Solve for a. a = [4]8 We can also write 8 as 2^3, so a = (2^3)^1/4 = 2^3/4. Step 3: Write the equation for g(x). y = (2^3/4)^x This can also be written as y = 8^x/4. The equation for g(x) is y = 8^x/4. 2.2.2 To determine the equation of g^-1(x), we first swap x and y in the equation for g(x). The question asks for the answer in the form x = .... Step 1: Start with the equation for g(x). y = 8^x/4 Step 2: Swap x and y to find the inverse relation. x = 8^y/4 The equation of g^-1(x) in the form x = ... is x = 8^y/4. 2.2.3 The function g(x) = 8^x/4 is an exponential function. Exponential functions of the form y = b^x have a horizontal asymptote at y=0. The inverse function g^-1(x) is a logarithmic function. The asymptote of an inverse function is found by swapping the variables of the original function's asymptote. Since g(x) has a horizontal asymptote at y=0, its inverse g^-1(x) will have a vertical asymptote at x=0. The equation for the asymptote of g^-1(x) is x = 0. Send me the next one 📸