Ways of Representing a Function

A function can be represented in several ways, and each way has its own scenario:

The Analytical (Formula) Method

Represent the functional relationship with a mathematical expression, such as $y = x^2 + 1$. This is the most commonly used method.

Advantages:

• Expresses the functional relationship precisely
• Makes mathematical computation convenient
• Makes the properties of the function clearly visible

Disadvantages:

• Some complicated functions are hard to express with a simple formula
• Not intuitive enough for piecewise functions

The Tabular Method

List the independent variable and the corresponding function values in a table.

The table below corresponds to the function $f(x)=(x-1)^2$:

Through the tabular method we can clearly see the function value corresponding to each independent variable, which makes it easy to analyse the pattern of change.

Advantages:

• Shows the concrete function values directly
• Makes it easy to look up the function value at a particular point
• Suited to handling discrete data

Disadvantages:

• Can only represent the function values at finitely many points
• Does not reveal the overall properties of the function
• Awkward to use when the table is large

The Graphical Method

Represent the functional relationship by a curve in the coordinate plane.

Advantages:

• Shows the overall shape of the function directly
• Makes it easy to observe the properties of the function (monotonicity, extrema, and so on)
• Suited to analysing the geometric features of the function

Disadvantages:

• Limited precision
• Makes precise numerical computation difficult
• The graph of a complicated function may be hard to draw

Exercises

Exercise 1

Given the function $f(x) = \begin{cases} x+1, & x \leq 0 \\ x^2, & x > 0 \end{cases}$, find the values of $f(-2)$, $f(0)$ and $f(3)$.

---

Summary

Symbols Used in This Article