3.1 Function Basics: Classification, Notation, and Evaluation

Quick Answer: A function pairs each input with exactly one output. On the Algebra 1 EOC, you classify functions as linear, quadratic, exponential, or absolute value from equations, graphs, and tables, then evaluate them using f(x) notation and interpret the result in context.

Functions vs. Relations and the Vertical Line Test

A relation is any set of ordered pairs. A function is a special relation in which each input (x-value) pairs with exactly one output (y-value). The input set is the domain; the output set is the range. The vertical line test determines whether a graph represents a function: if any vertical line drawn on the coordinate plane intersects the graph more than once, the relation is not a function. A circle fails this test because most vertical lines hit it twice, while a parabola, a straight line, and an exponential curve all pass. A single repeated y-value for two different x-values does not disqualify a relation from being a function; only a repeated x-value paired with two different y-values does.

Classifying Function Types from Equations, Graphs, and Tables

Classifying function types is a high-frequency EOC skill (MA.912.F.1.1). The four families you must recognize are linear, quadratic, exponential, and absolute value.

When classifying from a table, compute first differences (subtract consecutive y-values). If first differences are constant, the function is linear. If first differences are not constant, compute second differences (differences of the first differences). Constant second differences signal a quadratic. If neither is constant, check whether consecutive y-values share a constant ratio (each y-value multiplied by the same factor); that indicates exponential. An absolute value table shows symmetry: y-values repeat in reverse as you move away from the vertex x-value.

Worked example: A table lists x = 1, 2, 3, 4 and y = 3, 6, 12, 24. First differences are 3, 6, 12 (not constant). Second differences are 3, 6 (not constant). The ratio 6/3 = 2, 12/6 = 2, 24/12 = 2 is constant, so the function is exponential with growth factor 2.

Function Notation and Contextual Interpretation

Function notation (MA.912.F.1.2) uses f(x) to name the output associated with input x. The letter f is the function name; other letters (g, h, C, T) name different functions. Evaluating f(x) at a specific value means substituting that value for x and simplifying. For example, if f(x) = 3x² − 2x + 5, then f(4) = 3(16) − 8 + 5 = 45. On the EOC, function notation appears in context: if C(t) = 15t + 20 models the cost in dollars of t hours of tutoring, then C(3) = 65 means 3 hours of tutoring costs $65.

Interpreting in context is where students lose points. The EOC asks what a value like f(4) = 45 represents. The correct interpretation references the specific quantities: when the input is 4, the output is 45. If the function models temperature T(h) in degrees after h hours, then T(4) = 45 means after 4 hours the temperature is 45 degrees. Always pair the input quantity with its real-world meaning and the output quantity with its real-world meaning, and include units when the item provides them.

Common EOC traps:

• Confusing f(x) with multiplication. The notation f(x) is read 'f of x,' not 'f times x.' It is a label for the output.
• Forgetting that a function can have the same output for two different inputs. Only a repeated x-value paired with two different y-values breaks function status.
• Misclassifying a table with constant second differences as linear because the numbers change steadily. Always compute first and second differences before deciding.
• Assuming any curved graph is quadratic. Exponential curves have a horizontal asymptote and lack a vertex; quadratics have a vertex and axis of symmetry.
• Writing the domain of a function from a graph incorrectly. If the graph extends without bound in both horizontal directions, the domain is all real numbers, not a bounded interval.

EOC Traps, Graph Evaluation, and Piecewise Functions

Evaluating from a graph: When the EOC asks for f(2) given only a graph, locate x = 2 on the horizontal axis, move vertically to the curve, and read the y-coordinate. If the point is open at x = 2, the function has a discontinuity there and f(2) may be undefined or defined by a separate closed point elsewhere on the graph.

Piecewise evaluation: Some EOC items use piecewise functions, such as f(x) = { x + 1 if x < 0; x² if x ≥ 0 }. To evaluate, first determine which branch applies based on the input, then substitute. f(−3) uses the first branch: −3 + 1 = −2. f(3) uses the second branch: 9. A common error is applying the wrong branch because the student ignores the inequality sign attached to each piece. Check whether the input satisfies the branch condition (x < 0 or x ≥ 0) before substituting.