To find the slope of a linear function represented in a table of values, you need to identify two points from the table. Each point will be in the form of (x, y), where ‘x’ is the input value and ‘y’ is the output value. The slope (m) of a linear function is calculated using … Read more
Finding Solutions to the Equation To solve the equation 4 sin²(x) – 4 sin(x) + 1 = 0 within the interval [0, 2π], we can start by using a substitution method. Step 1: Substitute Let y = sin(x). Then, we can rewrite the equation as: 4y² – 4y + 1 = 0 Step 2: Factor … Read more
In set theory, the empty set, often denoted as {} or ∅, is fascinating because it is a subset of every set. To understand why this is the case, let’s break down a few concepts: Definition of a Subset: A set A is considered a subset of another set B if every element of A … Read more
To determine what a number is a percentage of another number, you can follow a simple mathematical formula. Let’s break it down step by step: Identify the percentage and the base number: First, clearly define the percentage you want to find and the base number (the number you think it might be a percentage of). … Read more
To find the equations of the tangent lines to the curve y = x^3 + x + 1 that are parallel to the line given by 2x – y = 4, we can follow these steps: Step 1: Find the slope of the given line First, we need to determine the slope of the line. … Read more
The rate of change between two points in mathematics is commonly represented by the concept of the slope or derivative, especially when dealing with a function. When we examine the interval from x = 0 to x = π/2, we can calculate the rate of change of a specific function, such as f(x) = sin(x), … Read more
Calculating the Area Enclosed by the Curves y = x and y = x3 To find the area enclosed between the curves of y = x and y = x3, we first need to determine the points where these two functions intersect. This involves solving for the values of x where: x = x3 By … Read more
To determine how deep the whale dives when it travels 200 meters at a 20-degree angle below the horizontal, we can utilize some basic trigonometry. Imagine a right triangle where: The hypotenuse (the side opposite the right angle) is the distance the whale travels underwater, which in this case is 200 meters. The angle with … Read more
To analyze the function f(x) = 3 cos(4x) + 2, we can break it down into its characteristics: amplitude, period, and midline. 1. Amplitude The amplitude of a cosine function can be determined from its coefficient in front of the cosine. In this case, the coefficient is 3. The amplitude is the distance from the … Read more
To find the value of f(g(5)), we first need to evaluate the inner function g(5). Given g(x) = 12x, we can substitute 5 for x: g(5) = 12 * 5 = 60 Now that we have g(5) = 60, we can substitute this value into the function f(x). Next, we evaluate f(60) using the function … Read more