Given the functions f(x)=x2-2x-3 and g(x)=x-3, where x ∈ R. Determine the value(s) of x, for which: (a) f(x) > g(x) (b) f(x).g(x) 0

1.2: Given functions: f(x) = x^2 - 2x - 3 and g(x) = x - 3. a) Determine the value(s) of x for which f(x) > g(x). Step 1: Substitute the expressions for f(x) and g(x) into the inequality. x^2 - 2x - 3 > x - 3 Step 2: Rearrange the inequality to have 0 on one side. x^2 - 2x - x - 3 + 3 > 0 x^2 - 3x > 0 Step 3: Factor the quadratic expression. x(x - 3) > 0 Step 4: Find the critical points by setting each factor to zero. x = 0 or x - 3 = 0 x = 3 The critical points are x = 0 and x = 3. Step 5: Determine the intervals where the inequality holds true. We test values in the intervals (-, 0), (0, 3), and (3, ). • For x < 0 (e.g., x = -1): (-1)(-1 - 3) = (-1)(-4) = 4 > 0. This interval is a solution. • For 0 < x < 3 (e.g., x = 1): (1)(1 - 3) = (1)(-2) = -2 < 0. This interval is not a solution. • For x > 3 (e.g., x = 4): (4)(4 - 3) = (4)(1) = 4 > 0. This interval is a solution. The solution is x < 0 or x > 3. x < 0 or x > 3 b) Determine the value(s) of x for which f(x) · g(x) ≤ 0. Step 1: Substitute the expressions for f(x) and g(x) into the inequality. (x^2 - 2x - 3)(x - 3) ≤ 0 Step 2: Factor the quadratic expression f(x) = x^2 - 2x - 3. x^2 - 2x - 3 = (x - 3)(x + 1) Step 3: Substitute the factored form of f(x) back into the inequality. (x - 3)(x + 1)(x - 3) ≤ 0 (x - 3)^2 (x + 1) ≤ 0 Step 4: Find the critical points by setting each factor to zero. x - 3 = 0 x = 3 x + 1 = 0 x = -1 The critical points are x = -1 and x = 3. Step 5: Determine the intervals where the inequality holds true. We consider the expression E = (x - 3)^2 (x + 1). Since (x - 3)^2 is always non-negative (i.e., (x - 3)^2 ≥ 0), the sign of E depends on the sign of (x + 1), unless (x-3)^2 = 0. • If x = 3, then E = (3 - 3)^2 (3 + 1) = 0 · 4 = 0. So x = 3 is a solution. • If x ≠ 3, then (x - 3)^2 > 0. For E ≤ 0, we must have (x + 1) ≤ 0. x + 1 ≤ 0 x ≤ -1 Combining these, the solution is x ≤ -1 or x = 3. The solution is x ≤ -1 or x = 3. x ≤ -1 or x = 3 Send me the next one 📸