Real-valued functions and their graphs

A real-valued function of a real variable is a function $f : D \to \mathbb{R}$ with $D \subseteq \mathbb{R}$ . The graph of $f$ is the set of points $(x, f(x))$ in the Cartesian plane. Recognising the graphs of standard functions is a Class-XI skill that you will use every day for the next several years.

The catalogue

1. Constant function

$f : \mathbb{R} \to \mathbb{R}$ , $f(x) = c$ . Graph: the horizontal line $y = c$ . Domain $\mathbb{R}$ , range $\{c\}$ .

2. Identity function

$f : \mathbb{R} \to \mathbb{R}$ , $f(x) = x$ . Graph: the line $y = x$ through the origin at $45^\circ$ . Domain $\mathbb{R}$ , range $\mathbb{R}$ .

3. Polynomial functions

$f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$ with real coefficients. Domain $\mathbb{R}$ . Range depends on degree.

Linear $f(x) = ax + b$ : a straight line.
Quadratic $f(x) = ax^2 + bx + c$ : a parabola, opening up if $a > 0$ , down if $a < 0$ . Vertex at $\left(-\frac{b}{2a},\ f\!\left(-\frac{b}{2a}\right)\right)$ .
Cubic $f(x) = x^3$ : passes through origin, increases everywhere, S-shape.

4. Rational functions

$f(x) = \dfrac{p(x)}{q(x)}$ where $p, q$ are polynomials. Domain is $\{x \in \mathbb{R} : q(x) \ne 0\}$ .

The simplest is the reciprocal $f(x) = \dfrac{1}{x}$ . Domain $\mathbb{R} \setminus \{0\}$ . Range $\mathbb{R} \setminus \{0\}$ . Graph: two branches (hyperbola), one in the first quadrant and one in the third.

5. Modulus (absolute value) function

$|x| = \begin{cases} x, & x \ge 0 \\ -x, & x < 0 \end{cases}$

Domain $\mathbb{R}$ , range $[0, \infty)$ . Graph: V-shape with vertex at origin; right side is $y = x$ , left side is $y = -x$ .

6. Signum function

$\text{sgn}(x) = \begin{cases} 1, & x > 0 \\ 0, & x = 0 \\ -1, & x < 0\end{cases}$

Domain $\mathbb{R}$ , range $\{-1, 0, 1\}$ . Graph: three pieces; two horizontal half-lines $y = 1$ and $y = -1$ , plus a single dot at origin. Note: $\text{sgn}(x) = \dfrac{|x|}{x}$ for $x \ne 0$ .

7. Greatest integer (floor) function

$\lfloor x \rfloor = \text{largest integer} \le x.$

Examples: $\lfloor 2.7 \rfloor = 2$ , $\lfloor -2.3 \rfloor = -3$ , $\lfloor 5 \rfloor = 5$ . Domain $\mathbb{R}$ , range $\mathbb{Z}$ . Graph: a staircase of horizontal segments, each of length $1$ , with closed dot on the left and open dot on the right of each step.

8. Fractional part function

$\{x\} = x - \lfloor x \rfloor$ . Domain $\mathbb{R}$ , range $[0, 1)$ . Graph: a sawtooth with period $1$ .

9. Square root function

$f(x) = \sqrt{x}$ , domain $[0, \infty)$ , range $[0, \infty)$ . Graph: a curve starting at the origin, increasing slowly. Note: by convention $\sqrt{x}$ denotes the non-negative root.

10. Exponential, logarithmic, trigonometric

You will meet $e^x$ , $\log x$ , $\sin x$ , $\cos x$ etc. as soon as we get to Chapter 3. They obey the same vocabulary: domain, range, graph.

Properties to remember

Even function: $f(-x) = f(x)$ . Graph is symmetric about the $y$ -axis. Examples: $x^2, |x|, \cos x$ .
Odd function: $f(-x) = -f(x)$ . Graph is symmetric about the origin. Examples: $x, x^3, \sin x, \text{sgn}(x)$ , $1/x$ .

For real functions, $f(x) = f(0) + (\text{odd part}) + (\text{even part})$ , but a deeper development of this waits for Class XII.

Worked examples

Example 1. Sketch $f(x) = |x - 1|$ .

Shift the V-graph of $|x|$ rightward by $1$ unit. Vertex at $(1, 0)$ . Right arm: $y = x - 1$ for $x \ge 1$ . Left arm: $y = 1 - x$ for $x < 1$ .

Example 2. Find the domain and range of $f(x) = \sqrt{x - 2}$ .

We need $x - 2 \ge 0$ , so $x \ge 2$ . Domain $= [2, \infty)$ . The output $\sqrt{x - 2}$ ranges over $[0, \infty)$ , so range $= [0, \infty)$ .

Example 3. Evaluate $\lfloor -1.5 \rfloor + \lfloor 2.7 \rfloor + \lfloor 3 \rfloor$ .

$\lfloor -1.5 \rfloor = -2$ (the largest integer $\le -1.5$ is $-2$ , not $-1$ ). $\lfloor 2.7 \rfloor = 2$ . $\lfloor 3 \rfloor = 3$ . Sum $= -2 + 2 + 3 = 3$ .

Example 4. Find the range of $f(x) = x^2 - 4x + 7$ .

Complete the square: $f(x) = (x - 2)^2 + 3 \ge 3$ , with equality at $x = 2$ . Range $= [3, \infty)$ .

Example 5 (harder). Sketch $f(x) = \text{sgn}(x) \cdot x^2$ .

For $x > 0$ : $f(x) = 1 \cdot x^2 = x^2$ (the right half of a parabola). For $x < 0$ : $f(x) = -x^2$ (the negative of $x^2$ , a downward curve in the third/fourth quadrant; here in the third). For $x = 0$ : $f(0) = 0$ . The graph is an S-shaped curve through the origin, looking like $x^3$ but flatter.

Note this function is odd: $f(-x) = \text{sgn}(-x)(-x)^2 = -\text{sgn}(x) x^2 = -f(x)$ .

Try it yourself

Sketch $f(x) = |x + 2|$ .
Find the domain and range of $f(x) = \frac{1}{x - 3}$ .
Compute $\lfloor 5.9 \rfloor, \lfloor -3.2 \rfloor, \lfloor 0 \rfloor, \lfloor -0.5 \rfloor$ .
Find the range of $f(x) = -x^2 + 4x + 1$ on $\mathbb{R}$ .
Sketch $f(x) = \text{sgn}(x - 1)$ .
Find domain of $f(x) = \sqrt{9 - x^2}$ .
Is $f(x) = x^4 - x^2$ even, odd, or neither?
Sketch the fractional part function $\{x\}$ on $[-2, 3]$ .
Find the range of $f(x) = \frac{x}{|x|}$ .
Compute $f(g(x))$ where $f(x) = \lfloor x \rfloor$ and $g(x) = x + 0.5$ .
Find domain of $f(x) = \sqrt{x + 3} + \frac{1}{\sqrt{5 - x}}$ .
Show that $f(x) = \sin x$ is odd and $f(x) = \cos x$ is even.

Pitfalls / Tricks

For negative real numbers, $\lfloor x \rfloor$ rounds toward $-\infty$ . So $\lfloor -1.5 \rfloor = -2$ , not $-1$ . A common exam trap.
$\sqrt{x}$ always denotes the non-negative root. The equation $x = y^2$ does not mean $y = \sqrt{x}$ in general.
For the reciprocal $1/x$ and modulus $|x|$ , never forget the special value at $x = 0$ .
Insight. Memorise the six core graphs , line, parabola, cubic, $1/x$ , $|x|$ , $\sqrt{x}$ . Most graphing questions reduce to shifting, scaling or reflecting one of these.