The cosine function has a specific structure that allows us to determine its characteristics easily. Let’s analyze the function y = 6cos(4x). 1. Domain The domain of a cosine function is all real numbers since the function is defined for every value of x. Therefore, for y = 6cos(4x), the domain is: Domain: (−∞, +∞) … Read more
The concept of an additive inverse refers to a number that, when added to the original number, results in zero. In other words, the additive inverse of a number ‘x’ is ‘-x’. For the number 6, the additive inverse would be: 6 + (-6) = 0 Thus, the additive inverse of 6 is -6. In … Read more
To determine whether the functions f and g are inverses, we need to verify that the compositions fg(x) and gf(x) equal x. In other words, we will check if: fg(x) = x gf(x) = x Given the functions: f(x) = 7x + 3 g(x) = 3x – 7 Step 1: Calculate fg(x) First, we substitute … Read more
An arithmetic sequence is defined by a formula that indicates how to generate its terms based on the variable n. In this case, the arithmetic sequence is given by the formula: an = 2 + 3n + 1 First, simplify the formula: an = 3n + 3 Now, let’s address the domain of n. The … Read more
To determine which statement is true given that x can take on the values of 1, 3, or 5, we need to analyze each of the statements presented in the context of the variable x. Let’s consider some potential statements: x > 0 x % 2 = 0 x < 6 x = 3 Now, ... Read more
How to Solve the Equation 8x² – 64x = 0 To solve the quadratic equation 8x² – 64x = 0, we can follow these steps: Step 1: Factor the Equation First, we can factor out the common term from both parts of the equation: 8x(x – 8) = 0 Step 2: Set Each Factor to … Read more
To find the linear approximation of the function f(x) = 1/x at a point, we typically use the formula: L(x) = f(a) + f'(a)(x – a) where: L(x) is the linear approximation f(a) is the value of the function at point a f'(a) is the derivative of the function at point a x is the … Read more
Understanding the Inequalities The given problem presents a system of inequalities that we need to solve. The inequalities are: 3x ≤ 12 100 ≥ 0 Breaking Down Each Inequality Let’s tackle each inequality one at a time: 1. Solving 3x ≤ 12 To isolate x, we need to divide both sides of the inequality by … Read more
The sequence given is 4, 9, 14, 19. To find the explicit formula for this pattern, let’s first observe the differences between consecutive terms: 9 – 4 = 5 14 – 9 = 5 19 – 14 = 5 The difference between consecutive terms is constant (5), indicating that this is an arithmetic sequence. In … Read more