Math Functions and Ordered Pairs Quiz

Function Representation

• The function $f(x) = (x + 1)^3 - (x - 1)^3$ can be simplified to represent a polynomial function.
• Through algebraic expansion, it resolves to $f(x) = 6x$.
• The function is linear, indicating a constant rate of change.

Intersection of Equations

• The equations $3x - y = 4$ and $x + y = 8$ can be solved simultaneously.
• By rearranging to express $y$, substituting, and solving yields the intersection point $(4, 4)$.

Equal Ordered Pairs

• In the pairs $(a + 2, 4)$ and $(5, 2a + b)$, both components must be equal.
• Setting $a + 2 = 5$ leads to $a = 3$.
• For the second component $4 = 2a + b$, substituting $a = 3$ results in $b = -2$.
• The values of $a$ and $b$ are $3$ and $-2$, respectively.

Identity Function Characteristics

• An identity function maps each element to itself; thus, the pairs must satisfy $a = 6$ and $b = 8$.
• This means that for the identity function to hold true, $a$ must equal $6$ and $b$ must equal $8$.

Prime Factorization of 1729

• The prime factorization of 1729 is $7^1 \times 13^1 \times 19^1$.
• The sum of the exponents in this factorization is $1 + 1 + 1 = 3$.