Math Analysis 1

Elementary functions and trigonometry: graphs, properties and transformations

Domain, symmetry and invertibility; powers, exponentials and logarithms; trigonometric functions and their inverses; sketching graphs with shifts and reflections.

ielettronica 19 October 2026 3 min read 0 comments

Before limits and derivatives you need to know the functions you work with: domain, sign, graph, symmetries, invertibility. This lesson reviews the elementary functions that appear in every first-year calculus exercise and the transformations that let you sketch new graphs without any computation.

Basic concepts

Domain: the set of values of $x$ for which the expression makes sense. Typical constraints are non-zero denominators, radicands of even-index roots $\ge 0$ , arguments of logarithms $> 0$ .
Even function: $f(-x) = f(x)$ , graph symmetric about the $y$ axis (like $x^2$ , $\cos x$ ). Odd: $f(-x) = -f(x)$ , symmetric about the origin (like $x^3$ , $\sin x$ ).
Monotone: increasing or decreasing on an interval. A strictly monotone function is invertible, and the graph of the inverse is the mirror image in the line $y = x$ .
Periodic with period $T$ : $f(x + T) = f(x)$ for every $x$ .

Powers and roots

$x^n$ with even $n$ is even and has its minimum at 0; with odd $n$ it is odd and increasing. Roots are the inverses: $\sqrt{x}$ is defined for $x \ge 0$ , $\sqrt[3]{x}$ on all of $\mathbb R$ . Powers with a real exponent $x^\alpha$ are defined for $x > 0$ as $e^{\alpha \ln x}$ .

Exponential and logarithm

$a^x$ with $a > 1$ is positive and increasing, with $0 < a < 1$ decreasing; it always passes through $(0, 1)$ . The logarithm $\log_a x$ is its inverse, defined for $x > 0$ and passing through $(1, 0)$ . Properties to know:

$\log_a(xy) = \log_a x + \log_a y, \quad \log_a\frac{x}{y} = \log_a x - \log_a y, \quad \log_a x^k = k\log_a x, \quad \log_a x = \frac{\ln x}{\ln a}.$

In engineering the logarithm is everywhere: decibels ( $20\log_{10}$ of a voltage ratio), Bode plots, the discharge of a capacitor following $e^{-t/\tau}$ .

Trigonometric functions

On the circle of radius 1, the point defined by the angle $x$ (in radians) has coordinates $(\cos x, \sin x)$ . The basic relations follow:

$\sin^2 x + \cos^2 x = 1$ , $\tan x = \dfrac{\sin x}{\cos x}$ ;
period $2\pi$ for sine and cosine, $\pi$ for tangent;
addition formulas: $\sin(a \pm b) = \sin a\cos b \pm \cos a \sin b$ , $\cos(a \pm b) = \cos a\cos b \mp \sin a\sin b$ ;
double angle: $\sin 2x = 2\sin x\cos x$ , $\cos 2x = \cos^2 x - \sin^2 x = 1 - 2\sin^2 x$ .

The related angles ( $\pi - x$ , $\pi + x$ , $\frac{\pi}{2} - x$ …) all follow from the symmetry of the circle: for instance $\sin(\pi - x) = \sin x$ and $\cos(\frac{\pi}{2} - x) = \sin x$ . The in-depth article on trigonometry collects them in a table.

Inverse trigonometric functions

Sine, cosine and tangent are not invertible on all of $\mathbb R$ , because they are periodic. They are restricted to an interval where they are monotone:

$\arcsin: [-1, 1] \to [-\frac{\pi}{2}, \frac{\pi}{2}]$ ;
$\arccos: [-1, 1] \to [0, \pi]$ ;
$\arctan: \mathbb R \to (-\frac{\pi}{2}, \frac{\pi}{2})$ .

Careful: $\arcsin(\sin x) = x$ only if $x \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ . For example $\arcsin(\sin \pi) = \arcsin 0 = 0$ , not $\pi$ .

Absolute value, integer part, piecewise functions

$|x|$ equals $x$ for $x \ge 0$ and $-x$ for $x < 0$ : a V-shaped graph. Piecewise functions model many signals (the step, the ramp, the rectangular pulse) and are the natural test bench for continuity and differentiability.

Sketching graphs with transformations

Starting from the graph of $f(x)$ you get, with no computation:

Function	Effect on the graph
$f(x) + k$	vertical shift by $k$
$f(x - h)$	horizontal shift by $h$ (to the right if $h > 0$ )
$-f(x)$	reflection in the $x$ axis
$f(-x)$	reflection in the $y$ axis
$\|f(x)\|$	the negative parts are flipped above the $x$ axis
$f(\|x\|)$	keep the part with $x \ge 0$ and mirror it to the left
$a f(x)$ , $f(bx)$	vertical stretch by $a$ , horizontal by $1/b$

Example: $y = 2 - e^{-x}$ comes from $e^x$ through a reflection in the $y$ axis ( $e^{-x}$ ), one in the $x$ axis ( $-e^{-x}$ ) and an upward shift by 2. The result starts at 1 for $x = 0$ and tends to 2: exactly the shape of the voltage on a capacitor charging towards 2 V.

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Basic concepts

Powers and roots

Exponential and logarithm

Trigonometric functions

Inverse trigonometric functions

Absolute value, integer part, piecewise functions

Sketching graphs with transformations

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