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.
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Fresh day asanda, let's solve. 2.1.1 v(x) is not a function. This is because it fails the vertical line test. For any x > 0, a vertical line drawn through the graph of v(x) intersects it at two distinct y-values (one positive and one negative), which violates the definition of a function. 2.1.2 To make v a function, we must restrict its range so that for each x-value, there is only one y-value. The conditions are: v(x) = sqrt(x) (where y 0) OR v(x) = -sqrt(x) (where y 0) 2.1.3 2.1.3.1 y for which w(x) < 0 The function w(x) = x. From the graph, w(x) is negative when 0 < x < 1. In this interval, the y-values of w(x) range from - up to (but not including) 0. Therefore, y < 0. 2.1.3.2 x for which w(x) -(7)/(10) The point A((49)/(100); -(7)/(10)) is on the graph of w(x). Since w(x) = x is an increasing function, for w(x) to be less than or equal to -(7)/(10), the corresponding x-values must be less than or equal to (49)/(100). Also, the domain of w(x) = x is x > 0. Therefore, 0 < x (49)/(100). 2.1.4 Assuming "5.1.2" is a typo and refers to "2.1.2", we consider the two conditions for v(x) to be a function: Case 1: If v(x) = sqrt(x) (with domain x 0 and range y 0) Step 1: Let y = sqrt(x). Step 2: Swap x and y: x = sqrt(y). Step 3: Square both sides to solve for y: x^2 = y. Step 4: The domain of v^-1(x) is the range of v(x), so x 0. Thus, v^-1(x) = x^2, for x 0. Case 2: If v(x) = -sqrt(x) (with domain x 0 and range y 0) Step 1: Let y = -sqrt(x). Step 2: Swap x and y: x = -sqrt(y). Step 3: Multiply by -1: -x = sqrt(y). Step 4: Square both sides to solve for y: (-x)^2 = y x^2 = y. Step 5: The domain of v^-1(x) is the range of v(x), so x 0. Thus, v^-1(x) = x^2, for x 0. 2.1.5 Assuming the question asks to "calculate the value of h(1)" (as "range of h(1)" is not standard terminology for a single point). Given h(x) = w(x) - sqrt(x). Step 1: Substitute x=1 into the expression for h(x). h(1) = w(1) - sqrt(1) Step 2: Evaluate w(1) using w(x) = x. w(1) = 1 = 0 Step 3: Evaluate sqrt(1). sqrt(1) = 1 Step 4: Substitute these values back into the expression for h(1). h(1) = 0 - 1 h(1) = -1 What's next? 📸