Unit notes

An input belongs to a function’s domain only if the original expression is defined there. A zero is an input producing output zero; an intercept is a point on an axis. A factor is a multiplicative piece of an expression, not a term separated by addition.

A function assigns exactly one output to each allowed input. Its domain may be restricted by an expression, such as a denominator, or by context, such as nonnegative time. A formula alone does not always describe the intended domain. Distinguish f(a), an output, from solving f(x) = a, which asks for inputs.

An average rate of change is the change in output divided by change in input over an interval. It is a secant slope and has output-units per input-unit. Positive values indicate net increase over the interval, but a function may rise and fall within that interval.

For equally spaced inputs, constant first differences support a linear model. Constant nonzero second differences support a quadratic model, and higher-order constant differences can indicate higher-degree polynomial structure. Equal spacing is essential to the simple finite-difference test.

Increasing average rates over successive equal intervals suggest upward curvature; decreasing rates suggest downward curvature. This differs from whether the function itself is increasing. A function can increase while its rate of increase decreases. Use interval-specific evidence instead of describing every upward-moving graph as “curving up.”

A real zero corresponds to a factor x − r. Odd multiplicity generally produces a sign change at the zero, while even multiplicity produces contact without crossing. Larger multiplicity can flatten the graph near the zero. Nonreal zeros occur in conjugate pairs for polynomials with real coefficients.

The leading term controls end behavior. Degree parity and leading-coefficient sign determine whether the ends point in the same or opposite directions. A polynomial of degree n has at most n − 1 turning points, but it need not achieve that maximum. Factoring, graphing, and numerical methods provide complementary information.

A rational function is a quotient of polynomials. Inputs making the original denominator zero are excluded. A common factor that cancels can produce a removable discontinuity, while a remaining denominator zero can produce a vertical asymptote. Cancellation simplifies values on the allowed domain; it does not restore excluded inputs.

End behavior depends on the quotient of leading terms or on polynomial division. A horizontal asymptote describes behavior far away, not a barrier the graph can never cross. Near a vertical asymptote, determine one-sided signs rather than assuming both sides go to positive infinity.

For y = a f(b(x − h)) + k, h and k shift the graph, a scales outputs, and b changes input scale. A negative a reflects vertically; a negative b reflects horizontally about the shifted input reference. Horizontal scaling uses a reciprocal factor, so f(2x) compresses x-coordinates by two.

Function composition feeds one output into another function. Its domain must satisfy both the inner function’s domain and the outer function’s restrictions on the resulting values. An inverse reverses a one-to-one input–output relationship; some functions need a restricted domain before an inverse function exists.

A polynomial or rational model can fit data well over one interval and behave unrealistically outside it. Interpret coefficients or rates only when the units and context support the interpretation. Extrapolating a population or cost model may violate physical constraints.

Residuals are observed minus predicted values. A systematic pattern suggests that the selected model misses structure. A small residual at one point is not enough to validate a model, and a high fit statistic does not prove the model’s mechanism is correct.

Factored form identifies candidate zeros and multiplicities. At a real zero of odd multiplicity, the sign changes; at one of even multiplicity, it does not. Higher multiplicity can flatten the graph near a zero, but it does not determine the entire graph. The leading term governs end behavior because it dominates lower-degree terms for sufficiently large input magnitude.

Combine local and global information. For a polynomial with positive leading coefficient and odd degree, the left end falls and the right end rises. Plot intercepts, determine signs between zeros, and check whether the proposed graph fits those conditions. A table of equal-step input values can reveal constant finite differences for a polynomial model, but measurement noise means exact patterns are uncommon in real data.

Factor numerator and denominator before canceling. A canceled denominator factor usually creates a hole at that input, whose missing height can be found from the simplified expression. An uncanceled factor in the denominator can create a vertical asymptote. The original denominator still controls the domain after simplification. Check the sign of the expression on each side rather than drawing both branches in the same direction by habit.

Long-run behavior depends on relative degrees. If the numerator’s degree is smaller, the output approaches zero. Equal degrees give the ratio of leading coefficients. If the numerator has higher degree, polynomial division separates a polynomial trend from a remainder that may approach zero. An asymptote describes limiting behavior; a graph can cross a horizontal or slant asymptote at finite inputs.

A model is a mathematical approximation with a domain of useful interpretation. Fit candidate models to the observed pattern, then examine residuals: observed output minus predicted output. Residuals scattered around zero without a systematic pattern support the model’s form; a curved residual pattern suggests a missing nonlinear relationship. A small numerical error on a tiny dataset is not enough to justify extrapolating far beyond it.

For transformations, distinguish changing the input from changing the output. In a f(b(x−h))+k, h shifts the input location, k shifts output, a changes vertical scale and possibly orientation, and b changes horizontal scale by a reciprocal factor. Track one known point from the original graph to the transformed graph to check the direction and scale before plotting everything.

A function assigns one output to each input in its domain. A formula, table, graph, and verbal rule reveal different information. A table shows selected values, not every value between them; a graph may hide a hole at its display resolution. State domain restrictions from denominators, even roots, and the context before simplifying.

Average rate of change over [a,b] is (f(b)−f(a))(b−a). Its units are output units per input unit, and it describes a secant slope. Equal output differences over equal input steps suggest a linear model; constant higher-order finite differences can indicate polynomial structure. A few matching data points do not prove a unique global model.

Transformations act on inputs and outputs differently. In f(x−h)+k, the graph shifts right by h and up by k. A horizontal scale acts inside the argument, so its geometric effect is reciprocal. Test a known point rather than memorizing a sign rule without checking it. Restrictions may change when a contextual model is transformed.

Factoring identifies zeros and multiplicities. At a zero of odd multiplicity, the sign changes; at an even multiplicity zero, it does not. Larger multiplicity can flatten the local graph, but the factor alone does not determine every turning point. The leading term governs end behavior as input magnitude becomes large.

A polynomial of degree n has at most n real zeros and at most n−1 turning points. These are upper bounds, not required counts. Complex zeros and repeated zeros affect the relationship between degree and distinct real intercepts. If coefficients are real, nonreal complex zeros occur in conjugate pairs.

To sketch responsibly, combine end behavior, intercepts, sign intervals, and values between important points. A graphing window can hide distant roots or make a steep curve look vertical. Use algebra to confirm features that a screen only suggests. A regression model fitted to observations should also be judged by residuals and contextual plausibility.

A rational expression is undefined wherever its original denominator is zero. Canceling a common factor produces an equivalent formula only on the original domain. The excluded input can become a removable discontinuity, while an uncanceled denominator zero can produce a vertical asymptote. Evaluate the reduced expression to locate the hole’s output when appropriate.

Near a vertical asymptote, inspect signs on each side to determine whether values grow positively or negatively without bound. For end behavior, compare degrees or use polynomial division. A horizontal asymptote describes far-away behavior and may be crossed at finite inputs. It is not automatically a barrier.

Solving a rational equation can introduce candidates that violate the original domain. Record restrictions, clear denominators carefully, solve, then check candidates in the original equation. For inequalities, use sign intervals separated by zeros and undefined points; multiplying by an expression of unknown sign can reverse an inequality unexpectedly.

Composition f(g(x)) applies g first and then f. The input must lie in g’s domain and its output must lie in f’s domain. In a contextual problem, follow the units through both stages. A mismatch can reveal that the composition order is wrong.

An inverse reverses input and output on a one-to-one domain. Restrict a non-one-to-one function when necessary, then solve y=f(x) for x with that restriction in mind. The inverse graph reflects across y=x. The reciprocal 1f(x) is a different operation.

Use an inverse when the output is given and the needed quantity is the original input. Check a composition such as f(f^-1(y))=y within the appropriate domain. A contextual inverse may have a limited range even when an algebraic formula appears to accept many values.