Write down the equations of the asymptotes of g. Determine the value(s) of x for which: g(x) = h(x), -4 < -2x-2/3 <= 0, g is increasing. Determine the equation of the line of symmetry of g with a negative x-intercept.

here are the solutions for the functions problem. 5.2.1 Write down the equations of the asymptotes of g. The given function is g(x) = (3)/(x) - 1. For a hyperbola of the form y = (k)/(x-p) + q: • The vertical asymptote is x = p. • The horizontal asymptote is y = q. Step 1: Identify p and q from the function g(x) = (3)/(x) - 1. Here, p=0 and q=-1. Step 2: Write down the equations of the asymptotes. The vertical asymptote is x=0. The horizontal asymptote is y=-1. x=0 and y=-1 5.2.2 Determine the value(s) of x for which: a) g(x) = h(x) Given g(x) = (3)/(x) - 1 and h(x) = -2x - 2. Setting them equal: (3)/(x) - 1 = -2x - 2 Multiplying by x (assuming x ≠ 0): 3 - x = -2x^2 - 2x Rearranging into a standard quadratic form ax^2 + bx + c = 0: 2x^2 + 2x - x + 3 = 0 2x^2 + x + 3 = 0 To find the solutions, we use the quadratic formula x = -b ± sqrt(b^2 - 4ac)2a. Here, a=2, b=1, c=3. x = -1 ± sqrt(1^2 - 4(2)(3))2(2) x = -1 ± sqrt(1 - 24)4 x = -1 ± sqrt(-23)4 Since the discriminant ( = -23) is negative, there are no real solutions for x. Note: The provided graph clearly shows two intersection points between g(x) and h(x). This indicates a likely typo in the definition of g(x). If g(x) was intended to be g(x) = -(3)/(x) - 1 (which matches the graph's appearance), the solutions would be: -(3)/(x) - 1 = -2x - 2 -3 - x = -2x^2 - 2x 2x^2 + 2x - x - 3 = 0 2x^2 + x - 3 = 0 Factoring the quadratic equation: (2x+3)(x-1) = 0 This gives two solutions: 2x+3 = 0 x = -(3)/(2) = -1.5 x-1 = 0 x = 1 Given the contradiction, I will provide the answer based on the algebraic definition as written, and then note the discrepancy. No real solutions b) -4 < (-2x-2)/(3) 0 This is a compound inequality. We solve each part separately. Step 1: Solve the left inequality: -4 < (-2x-2)/(3). Multiply both sides by 3: -12 < -2x - 2 Add 2 to both sides: -10 < -2x Divide by -2 and reverse the inequality sign: 5 > x or x < 5 Step 2: Solve the right inequality: (-2x-2)/(3) 0. Multiply both sides by 3: -2x - 2 0 Add 2 to both sides: -2x 2 Divide by -2 and reverse the inequality sign: x -1 Step 3: Combine the solutions from Step 1 and Step 2. We need x < 5 AND x -1. -1 x < 5 -1 x < 5 c) g is increasing. The given function is g(x) = (3)/(x) - 1. To determine where g(x) is increasing, we can examine its derivative or the behavior of the hyperbola. The derivative of g(x) is g'(x) = -(3)/(x^2). Since x^2 is always positive for x ≠ 0, g'(x) = -(3)/(positive number) will always be negative. This means g(x) is always decreasing for all x ≠ 0. Note: If we assume g(x) = -(3)/(x) - 1 (as suggested by the graph), then g'(x) = (3)/(x^2). In this case, g'(x) would always be positive, meaning g(x) is increasing for all x ≠ 0. Based on the written function, g(x) is never increasing. Never increasing 5.2.3 Determine the equation of the line of symmetry of g with a negative x-intercept. The lines of symmetry for a hyperbola of the form y = (k)/(x-p) + q pass through the intersection of its asymptotes (p, q) and have gradients of ± 1. From 5.2.1, the asymptotes of g(x) = (3)/(x) - 1 are x=0 and y=-1. Their intersection is (0, -1). Step 1: Write the equations of the lines of symmetry using the point-slope form y - y_1 = m(x - x_1). For m=1: y - (-1) = 1(x - 0) y + 1 = x y = x - 1 For m=-1: y - (-1) = -1(x - 0) y + 1 = -x y = -x - 1 Step 2: Determine which line has a negative x-intercept. For y = x - 1: Set y=0 0 = x - 1 x = 1. (Positive x-intercept) For y = -x - 1: Set y=0 0 = -x - 1 x = -1. (Negative x-intercept) The line of symmetry with a negative x-intercept is y = -x - 1. y = -x - 1 That's all for this one! Drop the next question.