A function is defined as a relation in which each input has exactly one output. This unique assignment distinguishes functions from other types of relations.

A polynomial is an algebraic expression that involves only non-negative integer exponents. Among the options given, 3x^2 + 2x - 1 is the polynomial expression because it contains only whole-number exponents.

The coefficient is the number that multiplies a variable in a term. It shows how many times the variable is taken and distinguishes the variable's scale in the term.

A linear equation is characterized by the fact that its graph forms a straight line. It involves variables raised only to the first power and does not include curves or exponents higher than one.

The degree of a polynomial is determined by the highest power of the variable present. In the polynomial 4x^3 + 2x^2 - x + 7, the term 4x^3 has the highest exponent, which is 3.

To solve 2x + 5 = 13, first subtract 5 from both sides to obtain 2x = 8, then divide by 2 to find x = 4. This step-by-step isolation of the variable ensures the correct solution.

Substituting x = 4 into the function f(x) = 2x + 3 gives f(4) = 2(4) + 3, which simplifies to 8 + 3 = 11. This function evaluation confirms the answer.

The equation a + b = b + a shows that the order of addition does not affect the sum, which is the definition of the commutative property. This property applies specifically to operations like addition and multiplication.

In Euclidean geometry, the sum of the interior angles of any triangle is always 180 degrees. This is a fundamental property used in many geometric proofs and calculations.

Adding 3 to both sides of the inequality x - 3 < 5 gives x < 8. This operation maintains the inequality's direction, resulting in the correct solution set.

The discriminant of a quadratic equation is given by b^2 - 4ac. This value is critical because it indicates the nature of the roots: if positive, there are two distinct real roots; if zero, one real root; and if negative, two complex roots.

The elimination method involves adding or subtracting entire equations to cancel out one of the variables. This method simplifies the system into a single equation that can be solved more easily.

The slope of a line indicates its steepness and direction, essentially representing the rate of change in y for each unit change in x. It is a key concept in understanding how linear functions behave.

A graph that passes the vertical line test confirms that for every x-value, there is only one corresponding y-value, meaning it represents a function. This is a standard method to determine if a relation is functional.

The associative property of addition states that the way in which numbers are grouped does not affect the final sum. This property allows flexibility when adding several numbers together.

To solve for y, add 2x to both sides to get 3y = 2x + 12, then divide the entire equation by 3. The resulting expression, y = (2/3)x + 4, represents y in terms of x.

Factoring the quadratic equation gives (x - 2)(x - 4) = 0. Setting each factor equal to zero yields the roots x = 2 and x = 4.

To find the x-intercepts, set f(x) = 0 which gives x^2 - 4 = 0. Factoring yields (x - 2)(x + 2) = 0, so x = 2 or x = -2 are the points where the graph crosses the x-axis.

Let the first integer be n, then the next consecutive integer is n + 1. The equation n + (n + 1) = 37 simplifies to 2n + 1 = 37, yielding n = 18. Thus, the integers are 18 and 19.

Dividing the area by the circumference gives (πr^2) / (2πr), which simplifies to r/2 after cancelling common terms. This ratio shows a direct relationship with the radius.