What is an affine function?

Definition

An affine function is a function f defined on R by an expression of the form:

f(x) = a*x + b

where a and b are two fixed real numbers. The number ‘a’ is called the slope, and ‘b’ is called the y-intercept.

Two important special cases

• If b = 0, the function is written as f(x) = a*x: this is called a linear function. Its graph always passes through the origin (0; 0).
• If a = 0, the function is written as f(x) = b: this is called a constant function. Its graph is a horizontal line.

Calculating an image and an antecedent

For the function f(x) = 3*x - 2:

• The image of 4 is f(4) = 3 × 4 − 2 = 10.
• The antecedent of 7 satisfies 3 × x − 2 = 7, so 3 × x = 9, hence x = 3.

Summary table

Form	Example	Name
f(x) = a*x + b, where a and b are non-zero	f(x) = 2*x + 5	affine
f(x) = a*x, where a is not equal to 0	f(x) = -3*x	linear
f(x) = b	f(x) = 4	constant

Common pitfall

Do not confuse an affine function with a linear function: a linear function IS an affine function (with b = 0), but an affine function is not always linear. Many pupils refer to any straight line as a ‘linear function’, which is incorrect as soon as b is not equal to 0.