Functions — NDA Mathematics

Formulas (1)

• Counting functions of a given type · Number of functions A → B
  $$\text{functions }A\to B=n^{m}\qquad \text{injections}={}^{n}P_{m}=\dfrac{n!}{(n-m)!}\qquad \text{bijections }(m=n)=n!\qquad \text{relations}=2^{mn}$$

Watch out for (3)

• Piecewise rules must agree at the boundary
  → Is it a function? Well-defined and the vertical-line test
• 'Onto' is not absolute — it depends on the codomain
  → One-one, onto, and bijective
• A non-monotone function usually fails one-one (and often onto)
  → One-one, onto, and bijective

Formulas (2)

• Even and odd functions · Even and odd tests
  $$f\text{ even}\iff f(-x)=f(x)\qquad f\text{ odd}\iff f(-x)=-f(x)$$
• Periodic functions and their period · Period after scaling the argument
  $$\text{period of }\sin(kx)=\dfrac{2\pi}{|k|}\qquad \text{period of }\tan(kx)=\dfrac{\pi}{|k|}\qquad \text{period of }f(ax+b)=\dfrac{T}{|a|}$$

Watch out for (4)

• ≥ 0 under a plain root, but > 0 when the root is a denominator
  → Finding the domain (roots, denominators, logs)
• Range is not the codomain
  → Finding the range
• $f(0)=0$ is necessary for odd, not sufficient
  → Even and odd functions
• $|x|$ is even, never odd
  → The modulus function and distance

Formulas (3)

• Composition of functions · Composition and its inverse
  $$(f\circ g)(x)=f(g(x))\qquad (f\circ g)^{-1}=g^{-1}\circ f^{-1}$$
• When do two linear functions commute? · Commuting condition for linear f, g
  $$f\circ g=g\circ f\iff b(c-1)=d(a-1)\ \ [f=ax+b,\ g=cx+d]$$
• Inverse of a function · Inverse of a linear function
  $$f(x)=ax+b\ (a\neq0)\ \Rightarrow\ f^{-1}(x)=\dfrac{x-b}{a}$$

Watch out for (3)

• $(fg)$ is a product, $(f\circ g)$ is a composition
  → Evaluating and combining functions
• Work inside-out, and keep the order
  → Composition of functions
• Inverse needs a bijection — and $f^{-1}\neq1/f$
  → Inverse of a function

Watch out for (2)

• Floor rounds DOWN, so negatives go further from zero
  → Definition, graph, and behaviour at integers
• [x] = n is an interval, not a single point
  → Floor equations and sums

Formulas (1)

• Multiplicative and additive forms · Signature functional-equation solutions
  $$f(xy)=f(x)f(y)\Rightarrow f(x)=x^{k}\qquad f(x+y)=f(x)f(y)\Rightarrow f(x)=a^{x}\qquad f(x+y)=f(x)+f(y)\Rightarrow f(x)=cx$$

Watch out for (2)

• One equation, two unknowns — make a second
  → Solving by substitution (x → 1/x, x → 1−x)
• Solve for the original variable before substituting
  → Undoing an argument shift