In short

A function can be defined in four standard ways. An explicit formula writes y directly in terms of x, like f(x) = 3x + 1. An implicit definition gives an equation linking x and y without solving for y, like x^2 + y^2 = 25. A piecewise function uses different formulae on different parts of the domain, like the absolute value. A parametric definition expresses both x and y in terms of a third variable (parameter), like the coordinates of a point tracing a circle. Each way suits different situations; the function itself is the same object regardless of how you describe it.

The Indian Railways timetable for the Rajdhani Express lists departure times for each station: Delhi at 16:55, Kanpur at 21:30, Allahabad at 23:45, and so on. Each station maps to exactly one departure time — that is a function. But nobody wrote a formula f(\text{station}) = \ldots to describe it. They wrote a table.

Now consider a different situation. A physics teacher says: the height of a ball thrown upward at 10 m/s is h(t) = 10t - 4.9t^2. Here the function is given by a formula — one expression that works for every value of t in the domain.

Both are valid functions. The difference is not in what a function is (one input, one output) but in how you specify the rule. There are four standard ways, and each is the natural choice in different situations.

Explicit formula

The most common way to define a function: write the output directly as an expression involving the input.

f(x) = 2x^2 - 3x + 7

This says: take the input x, square it, multiply by 2, subtract 3x, add 7. The output is completely determined by this one expression. For any specific x, you substitute and compute.

An explicit definition has the form y = (\text{expression in } x) — the output variable stands alone on one side. The expression on the right involves only the input variable and constants. Polynomials, rationals, trigonometric functions, exponentials — all the families of functions you will meet in class 11 and 12 are most often given this way.

Diagram showing an explicit function as an input-process-output machineA box labelled f with an arrow entering from the left carrying x and an arrow exiting to the right carrying 2x squared minus 3x plus 7. The box represents the function rule applied to the input to produce a single output. f rule: square, scale, add x 2x² − 3x + 7 explicit: output written directly in terms of input
An explicit function works like a machine: the input $x$ enters, the rule processes it, and a single output emerges. The formula $f(x) = 2x^2 - 3x + 7$ is a complete instruction set — you never need anything beyond the input to compute the output.

The strength of an explicit formula is that it is computable: give me any x in the domain, and I can hand you the output in one step. Its limitation is that not every function has a clean explicit formula — some are naturally described in other ways.

Implicit definition

Sometimes the relationship between x and y is expressed as an equation that involves both variables together, without isolating y on one side.

x^2 + y^3 - 6xy = 0

This equation links x and y, but it does not say "y = \ldots" — there is no clean way to solve for y in terms of x alone. The equation implicitly defines y as a function of x (at least locally, near points where certain conditions hold).

A familiar example: the equation of a circle, x^2 + y^2 = 25. This does not define y as a function of x over the whole circle (a circle fails the vertical line test). But if you restrict to the upper half — y = \sqrt{25 - x^2} — you get an explicit function. The implicit equation holds the information for both halves; extracting one half requires a choice.

A circle defined implicitly by x squared plus y squared equals 25 with the upper and lower halves labelledA circle of radius 5 centred at the origin. The upper semicircle is drawn as a solid curve and labelled y equals the positive square root of 25 minus x squared. The lower semicircle is drawn as a dashed curve and labelled y equals the negative square root. A vertical dashed line at x equals 3 hits the circle at two points, illustrating why the full circle is not a function. x y 5 −5 5 −5 y = +√(25 − x²) y = −√(25 − x²) x² + y² = 25 (implicit)
The circle $x^2 + y^2 = 25$ defined implicitly. The vertical line at $x = 3$ hits two points — one with $y > 0$ and one with $y < 0$ — so the full circle is not a function. The upper semicircle alone (solid) is a function; the lower semicircle alone (dashed) is another. The implicit equation packages both into one expression.

When is an implicit definition useful? When solving for y explicitly is either impossible or would produce an uglier expression than the original equation. In calculus, you will learn implicit differentiation — a technique to find \frac{dy}{dx} directly from the implicit equation, without ever solving for y. The implicit form is sometimes the more natural description of the relationship.

Piecewise functions

Some functions use different rules on different parts of the domain. The most familiar example is the absolute value:

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

For non-negative inputs, the output is x itself. For negative inputs, the output is -x (which is positive). The function is defined on all of \mathbb{R}, but it uses two different formulae depending on where the input falls.

A piecewise function is still one function — it has one domain, one codomain, and assigns one output to each input. The pieces must cover the entire domain (no input left out) and must not overlap in a conflicting way (if two pieces both apply at a boundary point, they must agree on the output).

Graph of the absolute value function as a piecewise functionA V-shaped graph on the x-y plane. For x less than 0, the graph follows y equals negative x, a line with slope negative 1. For x greater than or equal to 0, the graph follows y equals x, a line with slope positive 1. The two pieces meet at the origin, which has a filled dot. The left piece is labelled y equals negative x and the right piece is labelled y equals x. x y 2 −2 2 y = −x y = x
The absolute value $|x|$ is a piecewise function: $y = -x$ for $x < 0$ (left arm) and $y = x$ for $x \ge 0$ (right arm). The two pieces join at the origin with no gap and no ambiguity — the function is defined everywhere.

Here is a more interesting piecewise function:

g(x) = \begin{cases} x^2 & \text{if } x < 1 \\ 2x - 1 & \text{if } x \ge 1 \end{cases}

For x < 1, the rule is x^2 (a parabola). For x \ge 1, the rule is 2x - 1 (a straight line). At the boundary x = 1: the parabola gives 1^2 = 1, and the line gives 2(1) - 1 = 1. Both pieces agree at x = 1, so the function is continuous there (no jump). This agreement at the boundary is not automatic — you have to check it.

Graph of piecewise function: x squared for x less than 1 and 2x minus 1 for x at least 1A graph showing two pieces meeting at x equals 1. The left piece is a parabola y equals x squared from x equals negative 2 to x equals 1. The right piece is a line y equals 2x minus 1 from x equals 1 onward. They meet at the point (1, 1) with a filled dot, indicating the function is continuous there. x y 1 2 3 −1 1 2 y = x² y = 2x − 1
The piecewise function $g(x)$: a parabola for $x < 1$ and a straight line for $x \ge 1$. The two pieces meet smoothly at $(1, 1)$ — both rules give the same value there.

Piecewise functions appear constantly in real applications. Electricity tariffs, income tax slabs, speed limits that change at the city border — these are all piecewise-defined.

Parametric definition

Sometimes neither x nor y is naturally the "input." Instead, both are outputs of some third variable — called a parameter — that tracks progress along a curve.

A point moving around a circle of radius 5 can be described by:

x = 5\cos\theta, \qquad y = 5\sin\theta, \qquad \theta \in [0, 2\pi)

Here \theta is the parameter: it is the angle, and as \theta runs from 0 to 2\pi, the point (x, y) traces out the full circle. Neither x nor y is a function of the other over the full circle (the circle fails the vertical line test). But both x and y are functions of \theta.

A circle traced parametrically with angle thetaA circle centred at the origin with radius 5. A point on the circle at angle theta is marked with a red dot, with dashed lines dropped to the x-axis showing x equals 5 cos theta and to the y-axis showing y equals 5 sin theta. The angle theta is marked at the centre. x y θ 5 cos θ 5 sin θ (x, y) = (5 cos θ, 5 sin θ)
Parametric description of a circle. The parameter $\theta$ (the angle) determines both $x = 5\cos\theta$ and $y = 5\sin\theta$. As $\theta$ sweeps from $0$ to $2\pi$, the red point traces the full circle. Neither $x$ alone nor $y$ alone controls the motion — the parameter $\theta$ does.

Parametric definitions are the natural language for motion and curves. A projectile's path is given by x = v_0 t \cos\alpha and y = v_0 t \sin\alpha - \frac{1}{2}gt^2, where t is the parameter (time). You could eliminate t and write y as a function of x, but the parametric form keeps the physics visible: t tells you when the projectile is at each position, not just where.

When to use which

Method Best for Example
Explicit Computation, plotting, standard functions f(x) = x^3 - 2x
Implicit Curves that loop or fold, algebraic geometry x^2 + y^2 = 25
Piecewise Different rules in different regions, practical models Tax slabs, |x|
Parametric Motion, curves traced by a moving point x = \cos t, \; y = \sin t

These four ways are not mutually exclusive. A piecewise function is explicit within each piece. A parametric curve can sometimes be converted to implicit form by eliminating the parameter (x^2 + y^2 = 25 from x = 5\cos\theta, y = 5\sin\theta). The same function can wear different descriptions — the function itself is independent of how you write it down.

Two worked examples

Example 1: Write $f(x) = |2x - 3|$ as a piecewise function and sketch its graph

Step 1. Identify the boundary. The expression inside the absolute value is 2x - 3. It changes sign where 2x - 3 = 0, that is, x = \frac{3}{2}.

Why: the absolute value function switches between two rules at the point where its argument is zero. Finding that point is the first step in every piecewise conversion of an absolute value.

Step 2. Write the two pieces.

When x \ge \frac{3}{2}: the expression 2x - 3 \ge 0, so |2x - 3| = 2x - 3.

When x < \frac{3}{2}: the expression 2x - 3 < 0, so |2x - 3| = -(2x - 3) = 3 - 2x.

f(x) = \begin{cases} 3 - 2x & \text{if } x < \tfrac{3}{2} \\ 2x - 3 & \text{if } x \ge \tfrac{3}{2} \end{cases}

Why: removing the absolute value sign requires splitting into two cases — one where the argument is non-negative (leave it alone) and one where it is negative (negate it).

Step 3. Verify continuity at the boundary. At x = \frac{3}{2}: the left piece gives 3 - 2(\frac{3}{2}) = 3 - 3 = 0, and the right piece gives 2(\frac{3}{2}) - 3 = 3 - 3 = 0. Both agree — no jump.

Why: if the two pieces gave different values at the boundary, the function would have a jump discontinuity. Here they match, so the graph is one connected V-shape.

Step 4. Sketch. The left piece is a line with slope -2 and y-intercept 3. The right piece is a line with slope +2. They meet at (\frac{3}{2}, 0).

Result: f(x) = \begin{cases} 3 - 2x & \text{if } x < 3/2 \\ 2x - 3 & \text{if } x \ge 3/2 \end{cases}, with vertex at \left(\frac{3}{2}, 0\right).

Graph of f of x equals absolute value of 2x minus 3A V-shaped graph with vertex at (1.5, 0). The left arm has slope negative 2 and the right arm has slope positive 2. The y-intercept is at (0, 3). Grid lines mark x equals 1.5 and y equals 3. x y 3/2 3 3 y-intercept (0, 3) vertex (3/2, 0) slope = −2 slope = +2
The graph of $f(x) = |2x - 3|$: a V-shape with vertex at $(\frac{3}{2}, 0)$. The left arm has slope $-2$ (the piece $3 - 2x$) and the right arm has slope $+2$ (the piece $2x - 3$). The $y$-intercept at $(0, 3)$ matches $f(0) = |0 - 3| = 3$.

The V-shape is the signature of an absolute value function. The vertex sits at the point where the expression inside the bars equals zero, and the slopes on either side are equal in magnitude but opposite in sign.

Example 2: Eliminate the parameter from $x = 3\cos\theta$, $y = 3\sin\theta$ to find the implicit equation, and identify the curve

Step 1. Start with the parametric equations.

x = 3\cos\theta, \qquad y = 3\sin\theta

Why: to eliminate the parameter \theta, you need to find a relationship between x and y that does not involve \theta. The standard strategy for sine and cosine is to use the Pythagorean identity.

Step 2. Square both equations.

x^2 = 9\cos^2\theta, \qquad y^2 = 9\sin^2\theta

Step 3. Add the two equations.

x^2 + y^2 = 9\cos^2\theta + 9\sin^2\theta = 9(\cos^2\theta + \sin^2\theta) = 9 \cdot 1 = 9

Why: the identity \cos^2\theta + \sin^2\theta = 1 eliminates \theta completely, leaving a clean relationship between x and y.

Step 4. Identify the curve. The equation x^2 + y^2 = 9 is a circle centred at the origin with radius 3.

Result: The parametric curve x = 3\cos\theta, y = 3\sin\theta traces a circle of radius 3. Its implicit equation is x^2 + y^2 = 9.

Circle of radius 3 with parametric points marked at key anglesA circle centred at the origin with radius 3. Four points are marked at angles 0, pi over 2, pi, and 3 pi over 2, corresponding to the coordinates (3, 0), (0, 3), (negative 3, 0), and (0, negative 3). The equation x squared plus y squared equals 9 is labelled. x y θ = 0 → (3, 0) θ = π/2 → (0, 3) θ = π → (−3, 0) θ = 3π/2 → (0, −3) x² + y² = 9
The parametric curve $x = 3\cos\theta$, $y = 3\sin\theta$ traces a circle of radius $3$. Four key angles are marked: at $\theta = 0$, the point is at $(3, 0)$; at $\theta = \pi/2$, at $(0, 3)$; and so on. Eliminating $\theta$ gives the implicit equation $x^2 + y^2 = 9$.

The parametric form tells you how the point moves — counterclockwise, starting at (3, 0). The implicit form x^2 + y^2 = 9 tells you where all the points are but says nothing about the order of traversal. Both describe the same circle; each carries different information.

Common confusions

Going deeper

If you know the four ways to define a function and can convert between them where possible, you have the working toolkit for the next several chapters. The rest of this section is for readers who want to see how these definitions connect to deeper ideas.

Implicit function theorem (a preview)

When does an implicit equation F(x, y) = 0 actually define y as a function of x near a particular point? The answer is the implicit function theorem: if F is smooth enough and \frac{\partial F}{\partial y} \neq 0 at the point, then locally, y can be written as a function of x. The condition \frac{\partial F}{\partial y} \neq 0 ensures that the curve is not turning vertical at that point — exactly the condition under which the vertical line test passes locally.

For the circle x^2 + y^2 = 25 at the point (3, 4): \frac{\partial F}{\partial y} = 2y = 8 \neq 0. So near (3, 4), the circle does define y as a function of x (specifically, y = \sqrt{25 - x^2}). But at (5, 0): \frac{\partial F}{\partial y} = 0. There, the curve turns vertical, and you cannot write y as a function of x near that point.

Piecewise functions and Fourier series

Piecewise functions may seem like a mathematical compromise — stitching together different rules because no single formula works. But Fourier showed that any "reasonable" piecewise function can be written as a single (infinite) sum of sines and cosines. So in a deeper sense, every piecewise function does have a single formula — just not a finite one. This is the starting point of Fourier analysis, one of the most applicable branches of mathematics.

Parametric curves and velocity

When a curve is given parametrically as x(t), y(t), the parameter t usually represents time. The derivatives \frac{dx}{dt} and \frac{dy}{dt} are the horizontal and vertical components of velocity. The speed of the moving point is \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}. Parametric form is not just a way to describe a curve — it carries the dynamics of motion.

Where this leads next

You now know four ways to specify a function. The next articles explore what kinds of functions exist and how functions combine.