Inverse and square functions: variations and graphs

Square function: $f(x) = x^2$

Domain: $\mathbb{R}$ (any $x$).

Variations:

• Decreasing on $(-\infty, 0]$
• Increasing on $[0, +\infty)$
• Minimum at $0$: $f(0) = 0$.

Symmetry: $f(-x) = f(x)$ ⇒ even function (curve symmetric about the y-axis).

Graph: parabola with vertex $O(0, 0)$, opening upward.

Inverse function: $f(x) = \dfrac{1}{x}$

Domain: $\mathbb{R}^* = (-\infty, 0) \cup (0, +\infty)$ (forbidden: $x = 0$).

Variations:

• Decreasing on $(-\infty, 0)$
• Decreasing on $(0, +\infty)$

⚠️ NOT decreasing on $\mathbb{R}^*$ (interruption at 0).

Symmetry: $f(-x) = -f(x)$ ⇒ odd function (curve symmetric about the origin).

Graph: hyperbola with 2 branches, asymptotes $x = 0$ (vertical) and $y = 0$ (horizontal).

Comparison table

	Square $x^2$	Inverse $1/x$
Domain	$\mathbb{R}$	$\mathbb{R}^*$
Parity	Even	Odd
Symmetry	y-axis	Origin
Graph	Parabola	Hyperbola
Extremum	Min at 0	None

Common inequality solving

$x^2 < 4$ ⇒ $-2 < x < 2$ (inside the parabola).

$x^2 > 4$ ⇒ $x < -2$ or $x > 2$.

$\dfrac{1}{x} < 2$ for $x > 0$ ⇒ $x > \dfrac{1}{2}$ (careful with direction).

⚠️ $\dfrac{1}{x} < 2$ for $x < 0$: always true (negative inverse < 2).

Common mistakes

1. Saying $1/x$ is decreasing on $\mathbb{R}^*$ (false: decreasing on each interval separately).
2. Multiplying an inequality by $x$ without checking sign.
3. Forgetting the vertex at $0$ of the parabola.

?À toi de jouer

The function f(x) = x² is:

A.Even, minimum at 0B.Odd, minimum at 0C.Even, maximum at 0D.Neither even nor odd

Valider