Algebra of real functions

Real-valued functions can be added, subtracted, multiplied and divided , point by point , exactly like real numbers. This algebra of functions turns the set of real functions into something that almost behaves like the real number line, with one wrinkle: domains can shrink when you combine functions.

Definitions

Let $f, g$ be real-valued functions with domains $D_f$ and $D_g$ respectively, both contained in $\mathbb{R}$. For each $x$ in the common domain $D_f \cap D_g$:

• Sum: $(f + g)(x) = f(x) + g(x)$.
• Difference: $(f - g)(x) = f(x) - g(x)$.
• Product: $(fg)(x) = f(x) \cdot g(x)$.
• Scalar multiple: $(c f)(x) = c \cdot f(x)$, for any real $c$.

For the quotient $\dfrac{f}{g}$, we additionally require $g(x) \ne 0$: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}, \quad x \in D_f \cap D_g,\ g(x) \ne 0.$

The domain of any combination is the intersection of the individual domains, minus any points where the operation fails (division by zero).

Why "common domain" matters

If $f$ is defined only on $[0, \infty)$ (say $f(x) = \sqrt{x}$) and $g$ is defined only on $(-\infty, 1]$ (say $g(x) = \sqrt{1 - x}$), then $f + g$ is defined only where both are: on $[0, 1]$. Outside $[0, 1]$, at least one of the two terms is not real, so the sum is meaningless.

This rule, simple as it is, is the single biggest source of careless mistakes in JEE problems. Always intersect domains.

Algebraic laws

For functions $f, g, h$ on a common domain $D$:

• Commutative: $f + g = g + f$, $fg = gf$.
• Associative: $(f + g) + h = f + (g + h)$, $(fg)h = f(gh)$.
• Distributive: $f(g + h) = fg + fh$.
• Zero: the zero function $0(x) = 0$ satisfies $f + 0 = f$.
• One: the constant $1$ satisfies $f \cdot 1 = f$.

These all follow because the operations are defined pointwise and real numbers obey the same laws.

Worked examples

Example 1. Let $f(x) = x^2 + 1$ and $g(x) = 2x - 3$. Find $(f + g)(x), (f - g)(x), (fg)(x), (f/g)(x)$ and state domains.

$D_f = D_g = \mathbb{R}$.

• $(f + g)(x) = x^2 + 2x - 2$, domain $\mathbb{R}$.
• $(f - g)(x) = x^2 - 2x + 4$, domain $\mathbb{R}$.
• $(fg)(x) = (x^2 + 1)(2x - 3) = 2x^3 - 3x^2 + 2x - 3$, domain $\mathbb{R}$.
• $(f/g)(x) = \dfrac{x^2 + 1}{2x - 3}$, domain $\mathbb{R} \setminus \{3/2\}$.

Example 2. Let $f(x) = \sqrt{x}$ and $g(x) = \sqrt{1 - x}$. Find the domain of $(f + g)$ and $(fg)$.

$D_f = [0, \infty)$, $D_g = (-\infty, 1]$. Common domain $D_f \cap D_g = [0, 1]$, which is the domain of both $f + g$ and $fg$.

Example 3. Find $(fg)(x)$ and the domain of $f/g$, where $f(x) = x^2 - 4$ and $g(x) = x - 2$.

$(fg)(x) = (x^2 - 4)(x - 2) = (x - 2)^2 (x + 2)$.

$(f/g)(x) = \dfrac{x^2 - 4}{x - 2} = x + 2$ for $x \ne 2$. Even though the simplified expression is $x + 2$, the quotient's domain is $\mathbb{R} \setminus \{2\}$.

Example 4. Let $f(x) = x + 1, g(x) = 2x + 3$. Find a function $h$ such that $h + f = g$.

$h(x) = g(x) - f(x) = 2x + 3 - x - 1 = x + 2$.

Example 5 (harder). Let $f(x) = \dfrac{1}{x - 1}$ and $g(x) = \dfrac{1}{x + 1}$. Find $(f + g)(x)$ and state the domain.

$D_f = \mathbb{R} \setminus \{1\}$, $D_g = \mathbb{R} \setminus \{-1\}$. Common domain $= \mathbb{R} \setminus \{-1, 1\}$.

$(f + g)(x) = \dfrac{1}{x - 1} + \dfrac{1}{x + 1} = \dfrac{(x + 1) + (x - 1)}{(x - 1)(x + 1)} = \dfrac{2x}{x^2 - 1}$.

The simplified expression $\dfrac{2x}{x^2 - 1}$ has domain $\mathbb{R} \setminus \{-1, 1\}$ , matching what we computed.

Try it yourself

1. If $f(x) = x^2$ and $g(x) = x + 1$, find $(f + g)(x), (f - g)(x), (fg)(x), (f/g)(x)$, and state the domains.
2. If $f(x) = \sqrt{x - 1}$ and $g(x) = \sqrt{4 - x}$, find the domains of $f + g$ and $f/g$.
3. If $f(x) = x^3$ and $g(x) = x^2$, simplify $f/g$ and state its domain.
4. If $f(x) = |x|$ and $g(x) = x - 2$, find $(fg)(x)$ at $x = -1, 0, 1, 3$.
5. Show that if $f$ is even and $g$ is even, then $f + g, fg$ are even.
6. Show that if $f$ is odd and $g$ is odd, then $fg$ is even.
7. If $f(x) = 2x + 1$, find $f \cdot f$ and $f + f$. Are they equal? Why not?
8. Let $f(x) = \dfrac{1}{x}$. Find the domain of $f + f$ and of $f \cdot f$.
9. If $f(x) = \cos x$, $g(x) = \sin x$, what familiar function is $f^2 + g^2$?
10. If $f(x) = x^2 + 1$, find a function $g$ such that $fg(x) = x^4 - 1$.
11. If $f(x) = x + 2$ and $(f + g)(x) = 3x + 1$, find $g(x)$.
12. Find a function $f$ with $f(x) + f(-x) = 2x^2$ for all $x$.

Pitfalls / Tricks

• After simplifying a quotient, the domain may not extend. $\dfrac{x^2 - 4}{x - 2}$ simplifies to $x + 2$, but the domain of the quotient is still $\mathbb{R} \setminus \{2\}$. Functions are equal only when their domains are equal.
• $(f/g)$ requires $g(x) \ne 0$ , never forget.
• Adding two functions does not always give a function on the union of domains. Use the intersection.
• Insight. The algebra of functions makes the set of real functions a ring (you will meet this word in higher courses). Already in Class XI, recognising that $\sin^2 x + \cos^2 x = 1$ is a statement about functions , not just numbers , is a small but important step.