Reference functions: affine, square, inverse

The linear function

f(x) = a*x + b, where a and b are fixed real numbers.

• a is the slope: if a > 0, f is increasing on R; if a < 0, f is decreasing on R; if a = 0, f is constant.
• b is the y-intercept: the curve (a straight line) intersects the y-axis at the point (0; b).

Example: f(x) = -2x + 3 is decreasing (a = -2 < 0), and its line passes through the point (0; 3).

The square function

f(x) = x². Its graph is called a parabola. It is decreasing on ]–∞; 0], increasing on [0; +∞[, and has a minimum at 0 (f(0) = 0). It is symmetric about the y-axis.

The inverse function

f(x) = 1/x, defined on R excluding {0}. It is decreasing on ]-∞; 0[ and decreasing on ]0; +∞[ (note: decreasing on each of the two intervals separately).

Summary table

Function	Expression	Graph	Behaviour
Linear	a*x + b	straight line	increasing if a>0, decreasing if a<0
Quadratic	x^2	parabola	decreasing then increasing
Inverse	1/x	hyperbola	decreasing on each interval

Pitfall to avoid

For the inverse function, do not simply state that it is “decreasing on R excluding 0”: between a negative x close to 0 (very negative image) and a positive x close to 0 (very positive image), there is no continuous global decrease, due to the discontinuity at 0. You must always specify “on each of the two intervals separately”.