Branch of mathematics

Functions and analysis

Functions and their graphs, then limits, derivatives and integrals — the mathematics of change.

Topics in this branch

Intermediate
What a function is
A function is an unambiguous assignment: every argument gets exactly one value. Meet the f(x) notation, the domain and the range, the four ways to define a function, and the vertical line test on a graph.
Intermediate
Linear function
A linear function y = ax + b draws a straight line. Meet the meaning of the slope and the intercept, the zero, the formula of a line through two points, the general form, and the conditions for parallel and perpendicular lines.
Advanced
Quadratic function
A quadratic function f(x) = ax² + bx + c draws a parabola. Meet the vertex and the axis of symmetry, the three forms of the formula — general, canonical and factored — and how the sign of a decides between a minimum and a maximum.
Advanced
Reading properties from a graph
A graph shows the whole function at once: domain and range, zeros, sign, intervals of monotonicity and the extremum. Learn to read them off the drawing before you compute anything.
Advanced
Limit of a function
A limit says where the values of a function are heading as the argument closes in on a number — even when the function has no value at that number at all. It is the first idea of analysis and the ground the derivative stands on.
Advanced
The derivative
A derivative measures how fast a function changes at a point — it is the slope of the tangent to the graph. Meet the definition through the difference quotient and the six rules that compute it without taking a limit every time.
Advanced
What the derivative is for
The sign of the derivative says whether a function rises or falls, and its zeros point at the candidates for extrema. Learn to write the equation of a tangent, read a sign table and find a largest value.
Advanced
The integral
Integration is differentiation run backwards: we look for the function whose derivative is the one we started with. An indefinite integral gives a formula up to a constant, a definite one gives a number — and that number is the area under the curve.

Branch formulas

Branch: Functions and analysis

What a function is

  • A function from X into Y

    f ⁣:X→Yf \colon X \to Y

    assigns exactly one element of Y to every element of X

  • The value notation

    y=f(x)y = f(x)

    x is the argument, y the value of the function at it

  • Domain

    Df={x:f(x) exists}D_f = \{x : f(x) \text{ exists}\}

    the set of all arguments

  • Range

    Rf={f(x):x∈Df}R_f = \{f(x) : x \in D_f\}

    the set of all values the function takes

  • Zero of a function

    f(x0)=0f(x_0) = 0

    an argument whose value is zero

  • The graph

    graph of f={(x, f(x)):x∈Df}\text{graph of } f = \{(x,\, f(x)) : x \in D_f\}

    the set of points (argument, value)

Linear function

  • Slope–intercept form

    f(x)=ax+bf(x) = ax + b

    a — the slope, b — the intercept

  • Slope from two points

    a=y2−y1x2−x1a = \frac{y_2 - y_1}{x_2 - x_1}

    the rise divided by the run

  • The zero

    x0=−ba(a≠0)x_0 = -\frac{b}{a} \quad (a \neq 0)

    where the graph crosses the x-axis

  • The value at zero

    f(0)=bf(0) = b

    where the graph crosses the y-axis

  • General form

    Ax+By+C=0Ax + By + C = 0

    reducible to the slope–intercept form whenever B ≠ 0

  • Condition for parallel lines

    a1=a2a_1 = a_2

    parallel lines have equal slopes

  • Condition for perpendicular lines

    a1⋅a2=−1a_1 \cdot a_2 = -1

    hence a₂ = −1/a₁

Quadratic function

  • General form

    f(x)=ax2+bx+c(a≠0)f(x) = ax^2 + bx + c \quad (a \neq 0)

    the graph is a parabola

  • Coordinates of the vertex

    p=−b2a,q=−Δ4a=f(p)p = -\frac{b}{2a}, \quad q = -\frac{\Delta}{4a} = f(p)

    (p, q) — the turning point of the parabola

  • Canonical form

    f(x)=a(x−p)2+qf(x) = a(x - p)^2 + q

    the vertex is readable straight off it

  • Factored form

    f(x)=a(x−x1)(x−x2)f(x) = a(x - x_1)(x - x_2)

    exists only when Δ ≥ 0; shows the zeros

  • Axis of symmetry

    x=px = p

    the vertical line through the vertex

  • Range

    Rf=⟨q,+∞) for a>0R_f = \langle q, +\infty) \ \text{for } a > 0

    for a < 0 it is (−∞, q⟩

Reading properties from a graph

  • A zero

    f(x0)=0f(x_0) = 0

    where the graph crosses the x-axis

  • The sign of a function

    f(x)>0  ⟺  graph above the x-axisf(x) > 0 \iff \text{graph above the } x\text{-axis}

    below the axis the values are negative

  • Increasing function

    x1<x2  ⟹  f(x1)<f(x2)x_1 < x_2 \implies f(x_1) < f(x_2)

    on an interval where the graph rises

  • Decreasing function

    x1<x2  ⟹  f(x1)>f(x2)x_1 < x_2 \implies f(x_1) > f(x_2)

    on an interval where the graph falls

  • Minimum

    f(x)≥f(xmin⁡) for every xf(x) \geq f(x_{\min}) \ \text{for every } x

    the lowest point of the graph; a maximum works the same way

  • Domain and range

    Df→x-axis,Rf→y-axisD_f \to x\text{-axis}, \quad R_f \to y\text{-axis}

    the shadows of the graph on the two axes

Limit of a function

  • Limit of a function

    lim⁡x→x0f(x)=g\lim_{x \to x_0} f(x) = g

    the values f(x) close in on g as x closes in on x₀

  • Continuous function

    lim⁡x→x0f(x)=f(x0)\lim_{x \to x_0} f(x) = f(x_0)

    a polynomial is continuous, so the limit is plain substitution

  • The 0/0 indeterminate form

    lim⁡x→2x2−4x−2=lim⁡x→2(x+2)=4\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = \lim_{x \to 2} (x + 2) = 4

    cancel the common factor first, substitute afterwards

  • Equal degrees

    lim⁡x→∞anxn+…bnxn+…=anbn\lim_{x \to \infty} \frac{a_n x^n + \ldots}{b_n x^n + \ldots} = \frac{a_n}{b_n}

    the ratio of the leading coefficients

  • Numerator of lower degree

    lim⁡x→∞amxm+…bnxn+…=0(m<n)\lim_{x \to \infty} \frac{a_m x^m + \ldots}{b_n x^n + \ldots} = 0 \quad (m < n)

    the denominator grows faster, so the quotient dies away

  • One-sided limits

    lim⁡x→x0−f(x)=lim⁡x→x0+f(x)=g  ⟺  lim⁡x→x0f(x)=g\lim_{x \to x_0^-} f(x) = \lim_{x \to x_0^+} f(x) = g \iff \lim_{x \to x_0} f(x) = g

    a limit exists only when both sides agree

The derivative

  • Definition of the derivative

    f′(x0)=lim⁡h→0f(x0+h)−f(x0)hf'(x_0) = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}

    the limit of the difference quotient as the increment shrinks to zero

  • Power rule

    (xn)′=n⋅xn−1(x^n)' = n \cdot x^{n-1}

    the exponent steps in front of x and drops by one

  • Constants and sums

    (c⋅f)′=c⋅f′,(f+g)′=f′+g′(c \cdot f)' = c \cdot f', \qquad (f + g)' = f' + g'

    a constant factor passes through; a sum differentiates term by term

  • Product rule

    (f⋅g)′=f′g+fg′(f \cdot g)' = f' g + f g'

    the derivative of a product is NOT the product of the derivatives

  • Quotient rule

    (fg)′=f′g−fg′g2\left(\frac{f}{g}\right)' = \frac{f' g - f g'}{g^2}

    a difference on top — the order matters

  • Chain rule

    (f(g(x)))′=f′(g(x))⋅g′(x)\big(f(g(x))\big)' = f'(g(x)) \cdot g'(x)

    the outer derivative times the derivative of the inside

  • Root and reciprocal

    (x)′=12x,(1x)′=−1x2(\sqrt{x})' = \frac{1}{2\sqrt{x}}, \qquad \left(\frac{1}{x}\right)' = -\frac{1}{x^2}

    the same power rule with exponents ½ and −1

  • Trigonometric functions

    (sin⁡x)′=cos⁡x,(cos⁡x)′=−sin⁡x(\sin x)' = \cos x, \qquad (\cos x)' = -\sin x

    the cosine picks up a minus sign

What the derivative is for

  • Equation of the tangent

    y=f′(x0)(x−x0)+f(x0)y = f'(x_0)(x - x_0) + f(x_0)

    a line of slope f′(x₀) through the point of tangency

  • Increasing function

    f′(x)>0 on an interval  ⟹  f increasingf'(x) > 0 \ \text{on an interval} \implies f \ \text{increasing}

    a positive derivative is a tangent leaning upwards

  • Decreasing function

    f′(x)<0 on an interval  ⟹  f decreasingf'(x) < 0 \ \text{on an interval} \implies f \ \text{decreasing}

    a negative derivative is a tangent leaning downwards

  • Necessary condition for an extremum

    f′(x0)=0f'(x_0) = 0

    a critical point — a candidate, not yet an extremum

  • Sufficient condition (maximum)

    f′:+→− at x0  ⟹  maximum at x0f' : + \to - \ \text{at} \ x_0 \implies \text{maximum at} \ x_0

    the derivative must CHANGE sign from plus to minus

  • Sufficient condition (minimum)

    f′:−→+ at x0  ⟹  minimum at x0f' : - \to + \ \text{at} \ x_0 \implies \text{minimum at} \ x_0

    minus to plus — a trough instead of a crest

The integral

  • Antiderivative

    F′(x)=f(x)F'(x) = f(x)

    F is an antiderivative of f when differentiating F gives f

  • Indefinite integral

    ∫f(x) dx=F(x)+C\int f(x)\,dx = F(x) + C

    the result is a whole FAMILY of functions, hence the constant C

  • Power rule

    ∫xn dx=xn+1n+1+C(n≠−1)\int x^n\,dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)

    the exponent grows by one, then divide by the new exponent

  • Newton–Leibniz formula

    ∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\,dx = F(b) - F(a)

    the constant C cancels in the subtraction, so it never matters

  • Area under a curve

    f(x)≥0 on ⟨a;b⟩  ⟹  A=∫abf(x) dxf(x) \geq 0 \ \text{on} \ \langle a; b \rangle \implies A = \int_a^b f(x)\,dx

    the definite integral equals the area only above the axis

  • Area in general

    A=∫ab∣f(x)∣ dxA = \int_a^b |f(x)|\,dx

    where the curve dips below the axis, the area is computed piece by piece

  • Trigonometric functions

    ∫sin⁡x dx=−cos⁡x+C,∫cos⁡x dx=sin⁡x+C\int \sin x\,dx = -\cos x + C, \qquad \int \cos x\,dx = \sin x + C

    the minus sign changes sides compared with differentiation

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