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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Here are the solutions to the remaining questions: 2.2.3 Write down the domain for f(x). From the previous question, f(x) = x+2. This is a linear function. The domain of a linear function includes all real numbers. The domain for f(x) is x R. 2.2.4 Is f(x) continuous or discontinuous? Give a reason for your answer. The function f(x) is continuous. Reason: The graph of f(x)=x+2 is a straight line with no breaks, jumps, or holes. It can be drawn without lifting the pen. 2.3 Sketched below are the graphs of f(x) = (k)/(x) and g(x) = 2^x with coordinates of A (2, 4). 2.3.1 What is the mathematical name of the graph of g(x)? The graph of g(x)=2^x is an exponential \ function. 2.3.2 Determine the equation of the y-intercept of g(x). Step 1: To find the y-intercept, set x=0 in the function g(x)=2^x. g(0) = 2^0 g(0) = 1 The y-intercept of g(x) is (0, 1). 2.3.3 Is g(x) an increasing or decreasing function? Step 1: Observe the graph of g(x)=2^x. As the x-values increase, the y-values also increase. Alternatively, for an exponential function y=a^x, if the base a > 1, the function is increasing. Here, a=2, which is greater than 1. The function g(x) is an increasing \ function. 2.3.4 Determine the equation of f(x). Step 1: The function is given as f(x) = (k)/(x). Step 2: The graph shows that point A (2, 4) lies on f(x). Substitute x=2 and f(x)=4 into the equation to find k. 4 = (k)/(2) Step 3: Solve for k. k = 4 × 2 k = 8 Step 4: Substitute the value of k back into the function's equation. f(x) = (8)/(x) The equation of f(x) is f(x) = (8)/(x). 2.3.5 Determine the domain and the range of f(x). Step 1: For the function f(x) = (8)/(x), the denominator cannot be zero. The domain is all real numbers except x=0. Domain: \x R x ≠ 0\ or (-, 0) (0, ) Step 2: For the function f(x) = (8)/(x), the function can take any real value except zero. The range is all real numbers except y=0. Range: \y R y ≠ 0\ or (-, 0) (0, ) The domain is x R, x ≠ 0 and the range is y R, y ≠ 0. 3 done, 2 left today. You're making progress.