Graphing Complex Modulus Inequalities
Consider the inequality \(|2x – 3| – |x + 2| < 1\). This question requires you to: (a) Formulate the piecewise conditions by identifying the critical points where the expressions inside the absolute value change sign. (b) Solve the inequality on each resulting interval. (c) Graph the overall solution on a number line, indicating open […]
Piecewise Modeling of Absolute Value Functions from Data
A company collects data reflecting varying consumer satisfaction levels over time. The changes in behavior suggest that the relationship between the customer satisfaction rating, \(f(x)\), and time \(x\) in months can be modeled using a function involving absolute values. (a) Assuming experimental data is available in tabular form (not provided here), construct a piecewise model […]
Analyzing Modulus Inequalities Through Graph Separation
Consider the functions \(f(x)= x-2\) and \(g(x)= 3-x\). Solve the inequality \(|f(x)| < |g(x)|\) by completing the following steps: (a) Set up the piecewise definitions of \(|f(x)|\) and \(|g(x)|\) based on the critical points \(x=2\) and \(x=3\). (b) Analyze the graphs of \(y=|f(x)|\) and \(y=|g(x)|\) to determine their intersection point and relative positions. (c) Conclude […]
Combined Transformations of Modulus Functions
Consider the modulus function \(f(x) = |x|\). A transformed function is defined as \(g(x) = 3|x + 2| – 5\). (a) Describe the sequence of transformations applied to \(f(x)\) to obtain \(g(x)\). (b) Solve for the set of x-values for which \(g(x) < 4\).
Graphing Reflection of y = |f(x)|
Consider the function \(f(x) = x – 4\). Sketch the graph of \(y = f(x)\) and then the graph of \(y = |f(x)|\). Describe clearly how the portion of \(f(x)\) that is below the x-axis is transformed in \(y = |f(x)|\).