Find the following for the function \( f(x)=4 x^{2}+2 x-2 \). \( \begin{array}{llll}\begin{array}{lll}\text { (a) } f(0) & \text { (b) } f(4) & \text { (c) } f(-4) \\ \text { (e) }-f(x) & \text { (f) } f(x+3) & \text { (d) } f(-x) \\ \text { (g) } & \text { (h) } f(x+h)\end{array} \\ \text { (a) } f(0)=-2 \text { (Simplify your answer.) } \\ \text { (b) } f(4)=\square \text { (Simplify your answer.) }\end{array} \)

Let's dive into it and find the values step by step! To find \( f(4) \): 1. Substitute \( x = 4 \) into the function: \[ f(4) = 4(4)^2 + 2(4) - 2 \] \[ f(4) = 4(16) + 8 - 2 = 64 + 8 - 2 = 70. \] So, \( f(4) = 70 \). Next, for \( f(-4) \): 1. Substitute \( x = -4 \): \[ f(-4) = 4(-4)^2 + 2(-4) - 2 \] \[ f(-4) = 4(16) - 8 - 2 = 64 - 8 - 2 = 54. \] Thus, \( f(-4) = 54 \). Now, for \( f(-x) \): 1. Substitute \( -x \): \[ f(-x) = 4(-x)^2 + 2(-x) - 2 \] \[ = 4x^2 - 2x - 2. \] For \( -f(x) \): 1. First, find \( f(x) \): \[ f(x) = 4x^2 + 2x - 2. \] So: \[ -f(x) = - (4x^2 + 2x - 2) = -4x^2 - 2x + 2. \] To solve for \( f(x + 3) \): 1. Substitute \( x + 3 \): \[ f(x + 3) = 4(x + 3)^2 + 2(x + 3) - 2 \] \[ = 4(x^2 + 6x + 9) + 2x + 6 - 2 = 4x^2 + 24x + 36 + 2x + 4 = 4x^2 + 26x + 40. \] Lastly, for \( f(x + h) \): 1. Substitute \( x + h \): \[ f(x + h) = 4(x + h)^2 + 2(x + h) - 2 \] \[ = 4(x^2 + 2xh + h^2) + 2x + 2h - 2 = 4x^2 + 8xh + 4h^2 + 2x + 2h - 2. \] Here are the results: - (a) \( f(0) = -2 \) - (b) \( f(4) = 70 \) - (c) \( f(-4) = 54 \) - (d) \( f(-x) = 4x^2 - 2x - 2 \) - (e) \( -f(x) = -4x^2 - 2x + 2 \) - (f) \( f(x + 3) = 4x^2 + 26x + 40 \) - (h) \( f(x + h) = 4x^2 + 8xh + 4h^2 + 2x + 2h - 2 \) Ready for more math adventures? Let's keep the fun rolling!