This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Complete the table of values for x = 2^y. The given y values are -3, -2, -1, 0, 1, 2, 3. We calculate x for each y: If y = -3, x = 2^-3 = (1)/(8) If y = -2, x = 2^-2 = (1)/(4) If y = -1, x = 2^-1 = (1)/(2) If y = 0, x = 2^0 = 1 If y = 1, x = 2^1 = 2 If y = 2, x = 2^2 = 4 If y = 3, x = 2^3 = 8 The completed table of values is: |c|c| x & y \\ (1)/(8) & -3 \\ (1)/(4) & -2 \\ (1)/(2) & -1 \\ 1 & 0 \\ 2 & 1 \\ 4 & 2 \\ 8 & 3 \\ Step 2: Plot the graph of y = _2 x. Plot the points from the table: ((1)/(8), -3), ((1)/(4), -2), ((1)/(2), -1), (1, 0), (2, 1), (4, 2), (8, 3). Draw a smooth dashed curve through these points. This curve represents y = _2 x. Step 3: Plot the graph of y = 2^x. Since y = 2^x is the inverse of y = _2 x, its points are obtained by swapping the x and y coordinates from the table in Step 1. The points for y = 2^x are: (-3, (1)/(8)), (-2, (1)/(4)), (-1, (1)/(2)), (0, 1), (1, 2), (2, 4), (3, 8). Draw a smooth solid curve through these points. This curve represents y = 2^x. Step 4: Draw the line y = x. Draw a straight line passing through the origin (0,0) with a slope of 1. This line acts as the axis of symmetry between the graph of a function and its inverse. The final graph will show the exponential function y=2^x (solid curve), the logarithmic function y=_2 x (dashed curve), and the line y=x (typically a dashed or dotted line for reference), demonstrating their inverse relationship as mirror images across y=x. That's 2 down. 3 left today — send the next one.