Problem 116 - Precalculus Essentials

Question

What does it mean if a function \(f\) is increasing on an interval?

Step-by-Step Solution

Answer

A function \(f\) is increasing on an interval if for any two numbers \(x_1\) and \(x_2\) in the interval, \(f(x_2) > f(x_1)\) when \(x_2 > x_1\).

Step 1: Define Increasing Function

An increasing function is a type of function where the value of \(f(x)\) increases as \(x\) increases within a specified interval. That is, if any number \(x_2\) in the interval is greater than another number \(x_1\) in that interval, then \(f(x_2)\) is greater than \(f(x_1)\).

Step 2: Identify the relevant trigonometric identities

Based on the given expression or equation, identify which trigonometric identities (Pythagorean, double-angle, sum/difference, etc.) are applicable.

Step 3: Apply the identities and simplify

Apply the identified identities to transform the expression. Simplify step by step, combining like terms and reducing fractions where possible.

Step 4: Solve or evaluate

If solving an equation, isolate the trigonometric function and find the angle(s). If evaluating, compute the final numerical value.

Step 5: State the result

Express the final answer, including all solutions in the required domain if solving an equation.

Step 6: Conclude with the answer

A function \(f\) is increasing on an interval if for any two numbers \(x_1\) and \(x_2\) in the interval, \(f(x_2) > f(x_1)\) when \(x_2 > x_1\).