Functions

• How to Define and Typeset Your Own Function in LaTeX
  Define a mathematical function with standard notation, or create a reusable LaTeX command that typesets its name and input with correct spacing.
• How to Find a Maximal Domain for an Inverse Function
  To find a largest domain on which a function has an inverse, restrict it to an injective branch; the function itself determines which branches work.
• What Function Makes a Narrow, Bell-Shaped Peak?
  Use a Gaussian function for a narrow bell-shaped peak; adjust its center, height, and width, or use separate widths for an asymmetric curve.
• Is There an Injection Between the Natural and Rational Numbers?
  Yes: inclusion gives an injection from N to Q, and reduced fractions can be encoded as natural-number pairs to inject Q back into N.
• Is the Greatest Integer Function Monotonically Increasing?
  The greatest integer function is nondecreasing because x ≤ y implies floor(x) ≤ floor(y), but it is not strictly increasing.
• How to Make Any Function Repeat Periodically
  Turn a function on one interval into a periodic function using a modulo operation, with conditions for continuity and differentiability.
• When Is the Absolute Value of a Function Continuous?
  If f is continuous, then |f| is continuous. The converse is generally false, with a simple sign-changing counterexample.
• Why the Map f(x) = ax Is Bijective in a Group
  For a fixed element a in a group G, left multiplication f(x)=ax is bijective because its inverse is multiplication by a⁻¹.
• In What Sense Is a Function a Set?
  A function can be represented as a set of input-output ordered pairs; its graph records the mapping, while a triple can also record its codomain.
• How to Find the Support of a Function
  Find a function’s support by taking the closure of the points where it is nonzero; learn how endpoints, domains, and almost-everywhere definitions affect it.
• Examples of Additive-Only and Multiplicative-Only Functions
  See examples of functions that preserve sums but not products, or products but not sums, and learn when an additive example must be pathological.
• Even and Odd Fourier Series for f(x) = π − x on (0, π)
  Find the half-range cosine and sine Fourier series of f(x) = π − x on (0, π), with coefficient calculations, convergence details, and a check.
• Are the Positive Multiples of 17 and 11 in Bijection?
  The positive multiples of 17 and 11 are in bijection: match 17k with 11k for each positive integer k, and verify the map is one-to-one and onto.
• How Do You Convert a Discrete Function Into a Continuous One?
  A discrete function has no unique continuous conversion. Learn how to build a continuous extension with linear interpolation and when other methods are needed.
• How Do You Find the Slope of a Function?
  Find a function’s slope by differentiating it, then evaluate the derivative at the input you care about. The result is the tangent-line slope.
• How to Find Values of x Outside the Domain of h
  Find excluded x-values by checking denominators, even roots, logarithms, and restrictions in h; the formula for h is needed for a specific answer.
• How to Graph Functions Without a Calculator
  Graph a function by finding its domain, intercepts, symmetry, key points, turning points, and asymptotes, then sketching its overall behavior.
• How to Prove That a Function Is Well Defined
  To prove a function is well defined, check that every input has exactly one output in the codomain; for quotient sets, verify independence from the representative.
• Show That the Identity Function on Any Set Is a Bijection
  For any set A, the identity map sends each element to itself, making it injective and surjective—and therefore bijective, even when A is empty.