What is the input value for which the statement f(x) = g(x) is true?
To determine the input value for which the statement f(x) = g(x) holds true, you need to find the values of x that make the functions f and g equal to each other. This typically involves setting up the equation f(x) = g(x) and solving for x. For example, if f(x) = 2x + 3 […]
What is the Range of the Function g(x) = x^2 + 12?
The range of the function g(x) = x2 + 12 can be determined by analyzing its components. Since this is a quadratic function, it’s important to note the properties of the parabola it represents. The term x2 is always non-negative, meaning it has a minimum value of 0 when x = 0. Therefore, the smallest […]
Find the exact value of the real number y csc 1
To find the exact value of the real number y = csc(1), we start by recalling that cosecant is the reciprocal of the sine function. Thus, we have: csc(x) = 1/sin(x) Therefore, we can express y as: y = csc(1) = 1/sin(1) Next, we need to evaluate sin(1). The angle is in radians, and using […]
How do you find the values of all six trigonometric functions of a right triangle ABC where C is the right angle, given a = 7, b = 24, c = 25?
To find the values of all six trigonometric functions of the right triangle ABC with sides a = 7, b = 24, and c = 25, we start by identifying the relationships between the angles and the sides. In a right triangle: Sine of an angle (sin) is the ratio of the length of the […]
Find the vertex, focus, directrix, and focal width of the parabola x² = 12y
To analyze the parabola given by the equation x² = 12y, we need to identify several key components: the vertex, focus, directrix, and focal width. Firstly, we can rewrite the equation in the standard form of a parabola that opens upwards, which is (x – h)² = 4p(y – k). In this case, our equation […]
Write the Equation of a Parabola with Focus and Directrix
To write the equation of a parabola given its focus and directrix, we can use the definition of a parabola. A parabola is the set of all points (x, y) that are equidistant from a fixed point called the focus and a fixed line called the directrix. Assume that the focus is at the point […]
How to Use the Law of Sines to Find the Missing Side of a Triangle: What is Side b?
To find the missing side of a triangle using the Law of Sines, we need to understand the relationship between the sides and angles of a triangle. The Law of Sines states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is constant […]
Solve the Following System: 8x + 9y = 5 and 8x + 9y = 5
The given system of equations is: 8x + 9y = 5 8x + 9y = 5 Since both equations are identical, we can see that they represent the same line in the coordinate plane. This means that instead of a unique solution, we have infinitely many solutions. To express the solutions, we can rearrange the […]
How to Identify the Conic Section from the Equation x² + 4x + y² + 4y – 4 = 12?
To identify the conic section represented by the equation x² + 4x + y² + 4y – 4 = 12, we first rewrite the equation in a more standard form. Starting with: x² + 4x + y² + 4y – 4 = 12 We can rearrange this to: x² + 4x + y² + 4y […]
What is the future value of 500 one year from today if the interest rate is 6 percent?
To calculate the future value of an investment, we use the formula: Future Value (FV) = Present Value (PV) × (1 + r)^n Where: PV = Present Value (the initial amount of money) r = interest rate (as a decimal) n = number of years the money is invested or borrowed In this case: PV […]