91Ó°ÊÓ

Let's create a function f(x) that maps 3 to -4 and 8 to 1. We can do this by using a simple linear function in the form of f(x) = m * x + b, where m is the slope and b is the y-intercept. Since we have two points, (3, -4) and (8, 1), we can find the slope (m) by using the slope formula: m = \(\frac{y_{2} - y_{1}}{x_{2} - x_{1}}\) = \(\frac{1 - (-4)}{8 - 3}}\) = \(\frac{5}{5}\) = 1 Now that we know the slope, let's find the y-intercept (b) using one of the points, for example, (x,y) = (3, -4). -4 = 1(3) + b -4 = 3 + b b = -7 So, the first function f(x) is given by: f(x) = 1x -7

Since we already defined a linear function for the given domain and range, let's try a quadratic function for the second one. We can define g(x) in the form of g(x) = a(x-h)^2+k, where (h,k) is the vertex of the parabola. We still have the points (3, -4) and (8, 1) from the domain and range. We will use the average x-value to find the vertex of the parabola. h = \(\frac{3+8}{2}\) = 5.5 Now, let's find the value of k using the vertex equation. We can use the point (3, -4) to find k: -4 = a(3-5.5)^2 + k -4 = 6.25a + k We also need to use the other point (8, 1): 1 = a(8-5.5)^2 + k 1 = 6.25a + k Now we have a system of linear equations, and we can solve for a and k. Subtract the first equation from the second equation: 5 = 12.25a a = \(\frac{5}{12.25}\) Now plug the value of a back into the first equation to find k: -4 = 6.25\(\frac{5}{12.25}\) + k k = -1.795 We can now write the quadratic function g(x): g(x) = \(\frac{5}{12.25}\)(x-5.5)^2 - 1.795 The two different functions that satisfy the given domain and range are: 1. f(x) = x - 7 2. g(x) = \(\frac{5}{12.25}\)(x-5.5)^2 - 1.795