Algebraic Relationships and Functions
Domain overview
Algebraic Relationships and Functions asks you to represent, transform, and solve relationships using expressions, equations, inequalities, functions, and sequences. Items emphasize connections among symbolic, graphical, tabular, and verbal forms, and the grade 5–12 progression from proportional reasoning to quadratic and exponential models.
Core rules
- A function assigns exactly one output to each input; domain restrictions come from denominators (≠ 0), even roots (radicand ≥ 0), and logarithms (argument > 0).
- Linear form y = mx + b: m is rate of change, b is initial value; parallel lines share m, perpendicular slopes multiply to −1.
- Quadratic forms: standard y = ax^2 + bx + c gives vertex x = −b/(2a); factored form reveals zeros; vertex form reveals the maximum or minimum.
- Exponential growth and decay: y = a·b^x, with b > 1 growth and 0 < b < 1 decay.
- Solving equations: perform inverse operations, then check for extraneous solutions after squaring or multiplying by a variable expression.
- Systems: substitution, elimination, and graphing; the solution is the intersection, and no solution means parallel lines.
Worked example
Find the domain of f(x) = √(2x − 6)/(x − 4). The square root requires 2x − 6 ≥ 0, so x ≥ 3. The denominator requires x − 4 ≠ 0, so x ≠ 4. Combining gives [3, 4) ∪ (4, ∞). The reasoning is that each restriction is necessary; the domain is the intersection of the radicand condition and the denominator condition.
Traps
- Reporting x = 4 as included because the numerator is defined there.
- Reversing the inequality when multiplying by a negative.
- Confusing f(x − 2) with f(x) − 2; horizontal shifts act opposite the sign.
- Dropping the ± when solving x^2 = k.
- Assuming a relation is a function because it passes the vertical line test on a partial graph.