The inverse function for the function f(x) = x^2 is an important concept in mathematics, particularly in algebra and calculus. To find the inverse function, we follow a systematic approach. First, we start by replacing f(x) with y for easier manipulation: y = x^2 Next, we want to solve for x in terms of y. … Read more
To find an equivalent equation for the expression k 003 101k 245 181k, we first need to interpret the notation correctly. It seems to follow a format that includes variables (k) and numerical values. Here’s a breakdown of each part: k 003: This might be interpreted as the value of k multiplied by 3 or … Read more
Finding the X-Intercepts of the Equation The x-intercepts of a parabola are the points where the graph intersects the x-axis. This occurs when the value of y is equal to zero. To find the x-intercepts of the equation y = 2x² + 3x – 20, we need to set y to zero and solve for … Read more
To convert the quadratic equation f(x) = 8x² + 4x into vertex form, we need to complete the square. Vertex form is generally expressed as: f(x) = a(x – h)² + k where (h, k) is the vertex of the parabola, and a is a coefficient that affects the width and direction of the parabola. … Read more
Calculating the Area of the Office Space To find the area of the rectangular office space, we first need to understand the relationship between the width, length, and perimeter of a rectangle. We know the following: Width (W) = 72 feet Perimeter (P) = 410 feet Understanding the Perimeter Formula The formula for the perimeter … Read more
Finding the Orthocenter of Triangle ABC To find the orthocenter of triangle ABC with vertices A(0, 6), B(4, 6), and C(1, 3), we first need to understand what the orthocenter is. The orthocenter is the point where the three altitudes of a triangle intersect. Step 1: Sketch the Triangle Let’s plot the points: A: (0, … Read more
To rationalize the denominator of the expression sqrt(3) * sqrt(6) * sqrt(3) * sqrt(6), we will follow a systematic approach. Here’s how: Step 1: Multiply the Square Roots First, we can simplify the expression by combining the square roots: sqrt(3) * sqrt(6) * sqrt(3) * sqrt(6) = (sqrt(3) * sqrt(3)) * (sqrt(6) * sqrt(6)) This … Read more
To calculate the exact length of the curve defined by the function y = 3 – 2×3 from x = 0 to x = 1, we need to use the formula for arc length of a function given by: L = ∫ab √(1 + (dy/dx)2) dx Here, L represents the length of the curve, and … Read more
To calculate the future value of an investment, you can use the formula: Future Value (FV) = Present Value (PV) × (1 + r)^n Where: FV = Future Value PV = Present Value (initial investment) r = interest rate (as a decimal) n = number of years For this example, we have: PV = $100 … Read more
To express the relationship among the variables, we will start by understanding the concept of direct and inverse variation. When we say that y varies directly with x, it means that if x increases, y increases, and similarly, if x decreases, y decreases. In contrast, y varies inversely with z means that if z increases, … Read more