State whether v(x) is a function or not, motivate your answer. Write down conditions that will make v a function.

You're on a roll — here are the solutions for Question 2: 2.1.1 State whether v(x) is a function or not, motivate your answer. Step 1: Apply the vertical line test. A function must assign exactly one output value (y) for each input value (x). From the graph of v(x) = ±sqrt(x), for any x > 0, there are two corresponding y-values (one positive and one negative). For example, if x=1, y=± 1. Therefore, v(x) is not a function. Step 2: Provide the motivation. It fails the vertical line test because a vertical line drawn for any x > 0 would intersect the graph at two distinct points. Answer: v(x) is not a function because it fails the vertical line test. For a single x-value (where x>0), there are two corresponding y-values. 2.1.2 Write down conditions that will make v a function. Step 1: Restrict the range of v(x). To make v(x) a function, we must ensure that for each x, there is only one y. This can be done by choosing either the positive or the negative branch of sqrt(x). The conditions are: • v(x) = sqrt(x) for x 0 (restricting the range to y 0). • v(x) = -sqrt(x) for x 0 (restricting the range to y 0). Answer: v will be a function if its range is restricted to either y 0 (i.e., v(x) = sqrt(x)) or y 0 (i.e., v(x) = -sqrt(x)), with the domain x 0. 2.1.3 Determine all values of: 2.1.3.1 y for which w(x) < 0 Step 1: Analyze the graph of w(x) = x. The condition w(x) < 0 means that the graph of w(x) lies below the x-axis. From the graph, w(x) < 0 when 0 < x < 1. For these x values, the corresponding y values (which are w(x)) range from - (as x 0^+) up to 0 (as x 1^-). Answer: y < 0 (or y (-, 0)). 2.1.3.2 x for which w(x) -(7)/(10) Step 1: Use the given point A. We are given the point A((49)/(100), -(7)/(10)) on the graph of w(x). This means w((49)/(100)) = -(7)/(10). Step 2: Determine the interval for x. Since w(x) = x is an increasing function, if w(x) -(7)/(10), then x must be less than or equal to the x-value where w(x) = -(7)/(10). So, x (49)/(100). Also, the domain of w(x) = x is x > 0. Answer: 0 < x (49)/(100). 2.1.4 If a function v(x) is as determined in 2.1.2 write down the equation(s) of v^-1(x). Step 1: Consider the first case from 2.1.2: v(x) = sqrt(x). Let y = sqrt(x). To find the inverse, swap x and y: x = sqrt(y). Square both sides: x^2 = y. The domain of v^-1(x) is the range of v(x), which is y 0. So, x 0. Thus, v^-1(x) = x^2 for x 0. Step 2: Consider the second case from 2.1.2: v(x) = -sqrt(x). Let y = -sqrt(x). To find the inverse, swap x and y: x = -sqrt(y). Multiply by -1: -x = sqrt(y). Square both sides: (-x)^2 = y x^2 = y. The domain of v^-1(x) is the range of v(x), which is y 0. So, x 0. Thus, v^-1(x) = x^2 for x 0. Answer: The equations for v^-1(x) are v^-1(x) = x^2 for x 0 and v^-1(x) = x^2 for x 0. 2.1.5 If h(x) = w(x) - sqrt(x) where the range of sqrt(x) is (0, ), calculate the value of h(1). Step 1: Write down the function h(x). Given h(x) = w(x) - sqrt(x) and w(x) = x. So, h(x) = x - sqrt(x). Step 2: Calculate h(1). Substitute x=1 into the expression for h(x): h(1) = (1) - sqrt(1) Step 3: Evaluate the terms. We know that (1) = 0 and sqrt(1) = 1. h(1) = 0 - 1 h(1) = -1 Drop the next question.