Q.Draw the graph of the function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x^3$, $x \in \mathbb{R}$.

The key idea is that $f(x) = x^3$ is a cubic function with odd symmetry and a single inflection point at the origin.

Steps:

1. Symmetry: $f(-x) = (-x)^3 = -x^3 = -f(x)$, so the graph is symmetric about the origin (odd function).
2. Key points: $f(0) = 0$; as $x \to +\infty$, $f(x) \to +\infty$; as $x \to -\infty$, $f(x) \to -\infty$. …

The cubic function $f(x) = x^3$ is one of the simplest nonlinear functions you'll encounter, yet it has features that set it apart from both linear functions and quadratic parabolas. Let's build its graph from first principles.

Why this approach works

To draw any function's graph, you need to understand its behaviour — not just plot random points. For $x^3$, the key properties are:

• It's an odd function: $f(-x) = -f(x)$, so the graph is symmetric about the origin.
• It's strictly increasing everywhere: as $x$ grows, $y$ grows.
• It has no turning points (no local maxima or minima) — the derivative $f'(x) = 3x^2$ is zero only at $x=0$, but that's a point of inflection, not an extremum.

These three facts alone tell you the overall shape before you plot a single point.

Step-by-step construction

1. Identify the domain and range.

The function is defined for all real numbers, so the domain is $\mathbb{R}$. Since $x^3$ can take any real value (cube roots exist for all reals), the range is also $\mathbb{R}$. The graph extends infinitely in both directions.

2. Check symmetry.

$f(-x) = (-x)^3 = -x^3 = -f(x)$. This means the graph is symmetric about the origin: if $(a, b)$ lies on the graph, so does $(-a, -b)$. This halves the work — draw one side, then reflect.

3. Find intercepts.

Set $x = 0$: $f(0) = 0$. So the graph passes through the origin $(0,0)$. This is both the $x$-intercept and $y$-intercept — the only one, since $x^3 = 0$ only at $x=0$.

4. Analyse monotonicity (increasing/decreasing behaviour).

$f'(x) = 3x^2$. For any $x \neq 0$, $f'(x) > 0$, so the function is strictly increasing on $(-\infty, 0)$ and $(0, \infty)$. At $x=0$, the derivative is zero, but the function doesn't stop increasing — it just flattens momentarily.

5. Check concavity (curvature).

$f''(x) = 6x$.

• For $x < 0$, $f''(x) < 0$ → graph is concave down (curving downward).
• For $x > 0$, $f''(x) > 0$ → graph is concave up (curving upward).
• At $x = 0$, $f''(0) = 0$ and concavity changes — this confirms $(0,0)$ is a point of inflection.