In short

Seven basic functions — constant, identity, linear, quadratic, absolute value, square root, and cube root — appear so often that you should recognise each one's graph on sight. Each has a characteristic shape, a specific domain, and a specific range. Once you know these seven shapes, every transformation, composition, and piecewise function you meet in class 11 and 12 is built from pieces you already understand.

Walk into any rangoli competition during Diwali and you will notice that even the most elaborate patterns are built from a handful of basic shapes: circles, lines, dots, curves. A complex rangoli that fills an entire courtyard still decomposes into these simple elements repeated, rotated, and combined.

Function graphs work the same way. The functions you will meet in class 11 — rational functions, piecewise functions, transformed parabolas — are all built by shifting, stretching, reflecting, or combining a small set of basic shapes. This article catalogues those shapes. Know them well, and the more complex graphs will feel like variations on a familiar theme rather than new objects to memorise from scratch.

The constant function: f(x) = c

The simplest function assigns the same output to every input. Pick a constant c — say c = 3. No matter what x you feed in, the output is 3. The graph is a horizontal line at height c.

Domain: \mathbb{R} (every real number is a valid input). Range: \{c\} (only one output value ever appears).

Graph of the constant function f(x) = 3A horizontal line at y equals 3 on the x-y plane. The line extends in both directions, indicating the function is defined for all real numbers and always outputs 3. x y 3 1 −1 f(x) = 3
The constant function $f(x) = 3$. Every input produces the output $3$, so the graph is a flat horizontal line. Change the constant, and the line moves up or down — but it stays horizontal.

The constant function might look trivial, but it appears inside piecewise functions and as the horizontal asymptote that other functions approach.

The identity function: f(x) = x

The identity function returns whatever you give it. Input 5, output 5. Input -\pi, output -\pi. Its graph is the straight line through the origin at 45° — the line y = x.

Domain: \mathbb{R}. Range: \mathbb{R}.

Graph of the identity function f(x) = xA straight line passing through the origin at 45 degrees. Points (negative 2, negative 2), (0, 0), and (2, 2) are marked on the line. x y 2 −2 2 −2 (2, 2) (0, 0)
The identity function $f(x) = x$. The graph is the line $y = x$, passing through every point where the $x$- and $y$-coordinates are equal. It serves as the "mirror line" for reflecting graphs of inverse functions.

The identity function is the baseline against which other functions are compared. When you study inverse functions, the line y = x becomes the mirror line for reflection.

The linear function: f(x) = mx + b

Every straight line on the coordinate plane (that is not vertical) is the graph of a linear function f(x) = mx + b, where m is the slope and b is the y-intercept.

Domain: \mathbb{R}. Range: \mathbb{R} (when m \ne 0). If m = 0, the function is constant and the range is \{b\}.

Three linear functions with different slopesThree straight lines on the same axes. The line y equals 2x plus 1 has a steep positive slope. The line y equals x is the identity at 45 degrees. The line y equals negative x plus 3 has a negative slope, slanting downward to the right. x y 2 −2 2 y = 2x + 1 y = x y = −x + 3
Three linear functions. Changing the slope $m$ tilts the line; changing the intercept $b$ shifts it up or down. Every non-horizontal, non-vertical straight line is a linear function.

The slope m controls how steeply the line rises or falls. A positive m gives a line that rises to the right; a negative m gives a line that falls to the right. The identity function is the special case m = 1, b = 0.

The quadratic function: f(x) = x^2

The graph of f(x) = x^2 is a parabola — a U-shaped curve with its lowest point (vertex) at the origin. Every positive output is hit twice (once for a positive x, once for the corresponding negative x), and no negative output is ever produced.

Domain: \mathbb{R}. Range: [0, \infty).

Graph of f(x) = x squared, a parabola opening upwardA U-shaped parabola with vertex at the origin. Points (negative 2, 4), (negative 1, 1), (0, 0), (1, 1), and (2, 4) are marked. The curve is symmetric about the y-axis. x y 2 −2 1 −1 1 4 (2, 4) (0, 0)
The parabola $y = x^2$. It is symmetric about the $y$-axis: $f(-x) = f(x)$ for every $x$. The vertex at $(0,0)$ is the minimum point, so the range starts at $0$ and goes upward forever.

The symmetry f(-x) = (-x)^2 = x^2 = f(x) makes this an even function — its graph is a mirror image across the y-axis. The general quadratic f(x) = ax^2 + bx + c shifts and scales this basic shape, which you will explore in the article on quadratic expressions.

The absolute value function: f(x) = |x|

The absolute value of x measures how far x is from 0 on the number line, regardless of direction. It is defined by:

|x| = \begin{cases} x & \text{if } x \ge 0 \\ -x & \text{if } x < 0 \end{cases}

The graph is a V-shape with its vertex at the origin: two straight rays, one going up to the right (slope 1), the other going up to the left (slope -1).

Domain: \mathbb{R}. Range: [0, \infty).

Graph of f(x) = absolute value of x, a V-shapeA V-shaped graph with vertex at the origin. The left ray goes from (negative 3, 3) down to (0, 0), and the right ray goes from (0, 0) up to (3, 3). The graph is symmetric about the y-axis. x y 2 −2 2 4 (2, 2) (−2, 2)
The absolute value function $f(x) = |x|$. The sharp corner at the origin is where the two linear pieces meet. Like $x^2$, this is an even function: $|-x| = |x|$.

The absolute value function is closely related to the quadratic: both are even, both have range [0, \infty), and both "fold" negative inputs upward. The difference is the shape — a V versus a U — and the smoothness: |x| has a sharp corner at x = 0, while x^2 is smooth everywhere.

The square root function: f(x) = \sqrt{x}

The square root function takes a non-negative number and returns its non-negative square root. You cannot feed in a negative number (in the real number system), so the domain starts at 0.

Domain: [0, \infty). Range: [0, \infty).

Graph of f(x) = square root of xA curve starting at the origin and rising to the right, growing more and more slowly. Points (0,0), (1,1), (4,2), and (9,3) are marked. The curve exists only for x at least 0. x y 1 4 9 1 2 3 (9, 3) (4, 2)
The square root function $f(x) = \sqrt{x}$. It starts at the origin and rises, but its growth slows down: it takes $x = 9$ to reach $y = 3$. The curve is the upper half of a sideways parabola — the inverse of $y = x^2$ restricted to $x \ge 0$.

The square root curve rises steeply near x = 0 and then flattens out. Compare it with x^2: if you reflect the part of y = x^2 for x \ge 0 across the line y = x, you get y = \sqrt{x}. This is no coincidence — \sqrt{x} is the inverse function of x^2 on the restricted domain [0, \infty).

The cube root function: f(x) = \sqrt[3]{x}

Unlike the square root, the cube root is defined for all real numbers, including negatives. The cube root of -8 is -2 because (-2)^3 = -8. The graph passes through the origin and extends into both the third quadrant (negative x, negative y) and the first quadrant (positive x, positive y).

Domain: \mathbb{R}. Range: \mathbb{R}.

Graph of f(x) = cube root of xAn S-shaped curve passing through (negative 8, negative 2), (negative 1, negative 1), (0, 0), (1, 1), and (8, 2). The curve extends in both directions, defined for all real numbers. x y 1 −1 8 −8 2 −2 (8, 2) (−8, −2)
The cube root function $f(x) = \sqrt[3]{x}$. Unlike the square root, this curve lives on both sides of the origin. It is an odd function: $\sqrt[3]{-x} = -\sqrt[3]{x}$. The graph is symmetric about the origin (180° rotational symmetry).

The cube root function is an odd function: f(-x) = -f(x). Its graph has rotational symmetry about the origin — if you rotate it 180°, it looks the same. Compare this with the square root, which is only defined for x \ge 0 and has no such symmetry.

All seven at a glance

All seven basic functions plotted on the same axesSeven curves are plotted together: the constant function y equals 2 as a horizontal line, the identity y equals x, the parabola y equals x squared, the absolute value V-shape, the square root curve, the cube root curve, and a linear function y equals 2x plus 1. Each is labelled. x y 2 −2 2 −2 x² |x| y = x √x ∛x
Several basic functions on the same axes. The parabola $x^2$ (faint) opens upward. The V-shape of $|x|$ (grey) sits on top of it near the origin. The identity line $y = x$ (dashed) bisects the first quadrant. The square root $\sqrt{x}$ (red) and cube root $\sqrt[3]{x}$ (pink) both rise, but the square root is only defined for $x \ge 0$.

Two worked examples

Example 1: Sketch $f(x) = |x - 2|$ and state its domain and range

Step 1. Recognise the parent function. The expression |x - 2| is the absolute value function |x| shifted 2 units to the right.

Why: replacing x by x - 2 inside a function shifts its graph to the right by 2. This is a horizontal translation.

Step 2. Identify the vertex. The V-shape of |x| has its vertex at (0, 0). After shifting right by 2, the vertex moves to (2, 0).

Why: the expression |x - 2| equals 0 when x = 2, so the minimum point is at x = 2, y = 0.

Step 3. Draw the two rays. For x \ge 2, |x - 2| = x - 2 — a line with slope 1 starting at (2, 0). For x < 2, |x - 2| = -(x - 2) = 2 - x — a line with slope -1 starting at (2, 0).

Why: the absolute value "unfolds" at the point where its argument is 0. To the right, the argument is positive, so the absolute value equals the argument. To the left, the argument is negative, so the absolute value negates it.

Step 4. State the domain and range.

Domain: \mathbb{R} (every real number can be fed in). Range: [0, \infty) (the output is always non-negative, and every non-negative value is achieved).

Result: Domain = \mathbb{R}. Range = [0, \infty).

Graph of f(x) = |x − 2|, a V-shape with vertex at (2, 0)A V-shaped graph with vertex at (2, 0). The left ray has slope negative 1, going through (0, 2). The right ray has slope 1, going through (4, 2). The graph is labelled f(x) = |x − 2|. x y 0 2 4 6 2 4 vertex (2, 0) (0, 2) (4, 2)
The graph of $f(x) = |x - 2|$. The V-shape of $|x|$ has been shifted $2$ units to the right. The vertex sits at $(2, 0)$, confirming that the minimum output is $0$ and the range is $[0, \infty)$.

The graph confirms the algebra: the V-shape touches the x-axis at (2, 0) and rises on both sides, so no output is negative.

Example 2: Sketch $f(x) = \sqrt{x + 3}$ and find its domain and range

Step 1. Find the domain. The expression under the root must be non-negative: x + 3 \ge 0, so x \ge -3.

Why: the square root function only accepts non-negative inputs. The shift inside (x + 3) moves the starting point from 0 to -3.

Step 2. Recognise the transformation. The graph of \sqrt{x + 3} is the graph of \sqrt{x} shifted 3 units to the left.

Why: replacing x by x + 3 shifts the graph to the left by 3. The starting point moves from (0, 0) to (-3, 0).

Step 3. Plot key points. At x = -3: \sqrt{0} = 0. At x = -2: \sqrt{1} = 1. At x = 1: \sqrt{4} = 2. At x = 6: \sqrt{9} = 3.

Why: choosing values that make the expression under the root a perfect square gives clean output values for plotting.

Step 4. State the domain and range.

Domain: [-3, \infty). Range: [0, \infty) (same as the parent square root function, since a horizontal shift does not change the range).

Result: Domain = [-3, \infty). Range = [0, \infty).

Graph of f(x) = sqrt(x + 3), starting at (negative 3, 0)A square root curve starting at (negative 3, 0) and rising to the right through (negative 2, 1), (1, 2), and (6, 3). A dashed version of the basic sqrt(x) curve is shown faintly for comparison, starting at the origin. x y −3 −2 1 4 1 2 3 √x (−3, 0) (1, 2) √(x+3)
The graph of $f(x) = \sqrt{x+3}$ (solid red) compared with $f(x) = \sqrt{x}$ (dashed grey). The entire curve shifts $3$ units to the left. The starting point moves from $(0,0)$ to $(-3, 0)$, so the domain shifts from $[0, \infty)$ to $[-3, \infty)$, but the range stays $[0, \infty)$.

The dashed curve shows the parent function \sqrt{x} for comparison. The shift to the left changes the domain but not the range — the curve still rises from 0 upward.

Common confusions

Going deeper

The seven functions above are the ones you need for class 11. The ideas below connect them to two broader themes you will encounter later.

Even and odd functions

Three of the seven basic functions are even: f(-x) = f(x) for all x. These are f(x) = c (constant), f(x) = x^2, and f(x) = |x|. Their graphs are symmetric about the y-axis.

Two are odd: f(-x) = -f(x). These are f(x) = x (identity) and f(x) = \sqrt[3]{x} (cube root). Their graphs have 180° rotational symmetry about the origin.

The square root function is neither even nor odd — its domain is [0, \infty), so f(-x) is not even defined for x > 0.

A general linear function f(x) = mx + b is odd only when b = 0 (i.e., the line passes through the origin). With b \ne 0, it is neither even nor odd.

Transformations as a system

Once you know the seven basic shapes, every transformation follows a pattern. Given a parent function f(x):

These four moves — vertical shift, horizontal shift, vertical stretch/reflect, horizontal compress/reflect — generate every transformed basic function you will encounter. The parent shape determines the overall character; the transformation parameters fine-tune position and scale.

Where this leads next

You can now recognise the seven basic function shapes and state their domains and ranges on sight. Here is where each shape appears next.