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
Hey Jordan, good to see you again. Let's solve Question 3. Question 3: Given the function f(x) = 3+2x-x^2 defined on R, the set of real numbers. a) Using a scale of 2 \, cm for 1 \, unit on both axes, draw the graph of y = f(x) for range -2 x 4. Step 1: Create a table of values for y = 3+2x-x^2 for x from -2 to 4. • For x = -2: y = 3 + 2(-2) - (-2)^2 = 3 - 4 - 4 = -5 • For x = -1: y = 3 + 2(-1) - (-1)^2 = 3 - 2 - 1 = 0 • For x = 0: y = 3 + 2(0) - (0)^2 = 3 + 0 - 0 = 3 • For x = 1: y = 3 + 2(1) - (1)^2 = 3 + 2 - 1 = 4 • For x = 2: y = 3 + 2(2) - (2)^2 = 3 + 4 - 4 = 3 • For x = 3: y = 3 + 2(3) - (3)^2 = 3 + 6 - 9 = 0 • For x = 4: y = 3 + 2(4) - (4)^2 = 3 + 8 - 16 = -5 The table of values is: |c|c|c|c|c|c|c|c| x & -2 & -1 & 0 & 1 & 2 & 3 & 4 \\ y & -5 & 0 & 3 & 4 & 3 & 0 & -5 \\ Plot these points on a graph paper using the specified scale (2 \, cm for 1 \, unit on both axes) and draw a smooth curve through them. b) From the graph, find the solution of f(x) = 0. Step 2: To find the solution of f(x) = 0 from the graph, identify the x-intercepts (where the curve crosses the x-axis). From the table of values, we see that y=0 when x=-1 and x=3. Analytically, we solve 3+2x-x^2 = 0: x^2 - 2x - 3 = 0 (x-3)(x+1) = 0 x-3 = 0 or x+1 = 0 x = 3 or x = -1 The solutions are x = -1, x = 3. Drop the next question.