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Functions

YGK These Classifications of Mathematical Functions

QuestionAnswer
Which condition must a relation satisfy to be a function? Each input corresponds to exactly one output.
Which methods other than formulas can define a function? Tables or descriptions.
Which set contains all allowable input values of a function? The domain.
Which set lists all possible output values of a function? The codomain.
Which term refers to outputs a function actually attains? The image.
Why does NAQT avoid the word “range”? It can mean codomain or image.
Which type of expression consists of summed terms with nonnegative integer exponents? A polynomial.
Which part of a polynomial term is its numerical multiplier? The coefficient.
Which type of polynomial has exactly one term? Monomial.
Which type of polynomial has exactly two terms? Binomial.
Which type of polynomial has exactly three terms? Trinomial.
How is the degree of a single-variable polynomial determined? The highest exponent.
How is the degree of a multivariable polynomial determined? Largest sum of exponents in a term.
Which degree does a nonzero constant polynomial have? Degree 0.
Which degree classifies a polynomial as linear? Degree 1.
Which degree classifies a polynomial as quadratic? Degree 2.
Which degree classifies a polynomial as cubic? Degree 3.
What degree is assigned to the zero polynomial? Undefined or −∞.
Which theorem guarantees a complex root for every non-constant polynomial? Fundamental Theorem of Algebra.
Which theorem proves no general solution formula exists for quintics? Abel–Ruffini theorem.
Which field of math arose from studying polynomial solvability? Galois theory.
Which geometric shape represents the graph of a quadratic? Parabola.
Which coefficient determines whether a parabola opens upward or downward? The x² coefficient.
Which expression determines the number and type of quadratic roots? b² − 4ac.
What does a positive discriminant indicate? Two real roots.
What does a zero discriminant indicate? One real root with multiplicity 2.
What does a negative discriminant indicate? Two complex conjugate roots.
Which type of function is a ratio of two polynomials? Rational function.
Why must a rational function’s denominator be nonzero? Division by zero is undefined.
Why can every polynomial be considered a rational function? It can be written over 1.
Which graph feature shows where a function approaches but does not cross a line? An asymptote.
Which technique helps analyze rational-function asymptotes? Polynomial long division.
Which property defines a periodic function? f(x + p) = f(x).
Which quantity represents the repeat distance of a periodic graph? The period.
Which mathematical tool rewrites periodic functions using sines and cosines? Fourier series.
Which functions relate angles to triangle side ratios? Trigonometric functions.
Which geometric object is used to define trig functions algebraically? The unit circle.
Which six functions make up the basic trigonometric set? Sine, cosine, tangent, cosecant, secant, cotangent.
Which trig function equals sine divided by cosine? Tangent.
Which trig functions are reciprocals of sine, cosine, and tangent? Cosecant, secant, cotangent.
Which functions undo trigonometric functions? Inverse trigonometric functions.
Which prefix commonly names inverse trig functions? “Arc”.
Why must inverse trig functions have restricted domains? Trig functions are not bijective.
Which interval defines the domain of arcsin(x)? [−1, 1].
Which type of function never repeats output values? Injective function.
Which graphical test determines injectivity? Horizontal line test.
Which type of function attains every value in its codomain? Surjective function.
Which type of function is both injective and surjective? Bijective function.
Which property is required for a function to have an inverse? Bijectivity.
Which condition characterizes an even function? f(−x) = f(x).
Which symmetry does an even function’s graph exhibit? Reflection across the y-axis.
Which condition characterizes an odd function? f(−x) = −f(x).
Which symmetry does an odd function’s graph exhibit? 180° rotational symmetry about the origin.
Which function is both even and odd? The zero function.
Which functional form defines an exponential function? f(x) = bˣ.
Which restriction applies to the base of an exponential function? b > 0 and b ≠ 1.
Which specific exponential function uses Euler’s number? f(x) = eˣ.
Which calculus property makes exponentials unique? Their derivatives are proportional to themselves.
Which functions are inverses of exponential functions? Logarithmic functions.
Which base defines the natural logarithm? e.
Which base defines the common logarithm? 10.
Which base defines the binary logarithm? 2.
Which condition defines continuity at a point? The limit equals the function value.
Which graph features violate continuity? Holes, jumps, or asymptotes.
Which informal test describes continuity graphically? Drawing without lifting the pencil.
Which classes of functions are always continuous? Polynomials, sine, cosine, exponentials, logarithms, |x|.
Which condition defines differentiability at a point? Existence of the derivative.
Which property is stronger: continuity or differentiability? Differentiability.
Which graphical feature prevents differentiability? Corners or cusps.
Why is f(x) = |x| not differentiable at x = 0? It has a corner
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