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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Last free one today — make it count tomorrow, or type /upgrade for unlimited. 1.1 Given y = x 1.1.1 Sketch the graph Step 1: Identify key features of y = x. • The domain of y = x is x > 0. • The graph passes through the point (1, 0) because 1 = 0. • There is a vertical asymptote at x = 0 (the y-axis). • As x approaches 0 from the right, y approaches -. • As x approaches , y approaches . Step 2: Describe the sketch. To sketch the graph, draw a coordinate plane. Draw a dashed vertical line at x=0 (the y-axis) to represent the asymptote. Plot the point (1,0). Draw a smooth curve that starts from near the bottom of the y-axis (approaching x=0) and passes through (1,0), then continues upwards and to the right, increasing slowly. 1.1.2 State the range of the graph mentioned. Step 1: Determine the possible y-values for y = x. As x varies over its domain (0, ), the value of x can take any real number. Step 2: State the range. The range of y = x is all real numbers. (-, ) or \y y R\ 1.1.3 Is the graph continuous or discontinuous? Step 1: Analyze the behavior of the function over its domain. A function is continuous if its graph can be drawn without lifting the pen. The natural logarithm function has no breaks, jumps, or holes within its domain x > 0. Step 2: Conclude continuity. The graph of y = x is continuous over its domain. 1.1.4 Is the graph a function or a relation? Step 1: Apply the vertical line test. For every valid input x in the domain (x > 0), there is only one corresponding output y. A vertical line drawn anywhere in the domain x > 0 will intersect the graph at most once. Step 2: Conclude function or relation. The graph of y = x is a function. 1.1.5 Write down the equation of the inverse of the graph mentioned in question 1.1 Step 1: Swap x and y in the original equation. Given y = x. Swap variables: x = y. Step 2: Solve for y. To isolate y, convert the logarithmic equation to an exponential equation. The natural logarithm y is _e y. So, x = _e y implies y = e^x. Step 3: State the inverse equation. The equation of the inverse is: y = e^x