Mathematical Modeling with Functions Quiz

Linear functions have a constant rate of change, and this equation is in the form y = mx + b, where m is the slope and b is the y-intercept.

The vertex of a quadratic function in the form ax^2 + bx + c is given by (-b/2a, f(-b/2a)). In this case, the vertex is (-2/3, -5/3).

The range of the reciprocal function 1/x is all real numbers except 0, represented as (-∞, 0) ∪ (0, ∞).

The composite function (f ⨬ g)(x) is obtained by substituting g(x) into f(x), resulting in 8x - 1.

The absolute value function is defined for all real numbers, so its domain is (-∞, ∞).

To find the inverse function, swap x and y and solve for y. The given expression represents the inverse of the original function.

The square root function is defined only for non-negative values under the radical. Solving 4 - x^2 ≥ 0 gives the domain of (-2, 2).

A piecewise function is defined by different expressions over distinct intervals or sets of inputs.

The logarithm of a positive number is defined, so the domain of this logarithmic function is (0, ∞).

The roots are the values of x that make the polynomial equal to zero. In this case, the roots are -1, 2, and 1.

Integrate each term of the polynomial to get x^3 + C, where C is the constant of integration.

An odd function satisfies f(-x) = -f(x). The sine function has this property, making it an odd function.

Simplify the expression to (x + 1) and then substitute x = 1 to find that the limit is 2.

The Maclaurin series expansion of sin(x) is x - x^3/6, where each term is derived from the function's derivatives at x = 0.

Apply the chain rule to find the derivative of e^(2x), resulting in 2e^(2x).

The logarithmic form expresses the exponent as the logarithm base 2 of the result. In this case, log₂(256) = 8.

Use L'Hôpital's Rule or simplify the expression to recognize that the limit is e.