Lesson 5 of 10 · 8 min
Identity, constant and polynomial functions
NCERT §2.4.1
At every kilometre stone the bus passes, the stone's number is exactly the bus's position x. Meanwhile the class fund charges every student the same ₹50, however far they ride. Two very different rules, and both are functions.
The lesson in notes
In short
The identity function f : R → R, f(x) = x, sends each number to itself. Its domain and range are both R, and its graph is the straight line y = x through the origin, making 45° with the x-axis.
A constant function f : R → R, f(x) = c, sends every x to the same number c. Its domain is R, its range is the single-element set {c}, and its graph is a line parallel to the x-axis at height c.
A polynomial function has the form f(x) = a₀ + a₁x + a₂x² + … + aₙxⁿ, where n is a non-negative integer and a₀, a₁, …, aₙ are real numbers. x³ − 4x + 1 is one; √x + x and x + 1/x are not, because they contain a power of x that is not a non-negative integer.
f(x) = x² has domain R and range [0, ∞), since squares are never negative. Its graph is a U-shaped curve (parabola), symmetric about the y-axis, with lowest point at the origin: f(−3) = f(3) = 9.
f(x) = x³ has domain R and range R. Its graph rises through the origin, and it keeps the sign of x: f(−2) = −8 and f(2) = 8.
A function f(x) = mx + c with constants m and c is called a linear function; its graph is a straight line. The identity (m = 1, c = 0) and constant functions (m = 0) are special cases.