If $f$ and $g$ are two real valued functions defined as $f(x) = 2x + 1$, $g(x) = x^2 + 1$, then find. (i) $f + g$ (ii) $f - g$ (iii) $fg$ (iv) $\dfrac{f}{g}$

If f and g are two real valued functions defined as f(x) = 2x + 1, g(x) = x² + 1, then find. (i) f + g (ii) f

NCERT Exemplar · Q13

f

(i)

f + g

(ii)

f - g

(iii)

fg

(iv)

\dfrac{f}{g}

Sikkim CbseShort· 2mImportance★★★★★est

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Concept understanding — Function Operations

Function Operations: Combining Machines

Think of a function as a machine. You feed it an input (say, a number $x$ ), it does something, and out comes an output $f(x)$ . Now imagine you have two such machines, $f$ and $g$ . Function operations are simply ways to hook these machines together — to add, subtract, multiply, or divide their outputs, or to feed one machine's output into the other.

The core idea is simple: if you can do arithmetic with numbers, you can do arithmetic with functions. The only catch is that both functions must be "ready to work" on the same input at the same time.

The Four Arithmetic Operations

Let $f$ and $g$ be two functions. For any input $x$ that belongs to both their domains (the set of numbers each can accept), we define:

Operation	Notation	What it means
Sum	$(f+g)(x)$	$f(x) + g(x)$
Difference	$(f-g)(x)$	$f(x) - g(x)$
Product	$(f \cdot g)(x)$	$f(x) \cdot g(x)$
Quotient	$\left(\frac{f}{g}\right)(x)$	$\frac{f(x)}{g(x)}$ , provided $g(x) \neq 0$

Note

The domain of the new function is the intersection of the domains of $f$ and $g$ — the numbers both machines can handle. For the quotient, you must also exclude any $x$ where $g(x) = 0$ , because division by zero is undefined.

Example. Let $f(x) = \sqrt{x}$ (domain: $x \geq 0$ ) and $g(x) = x - 1$ (domain: all real numbers). Then:

$(f+g)(x) = \sqrt{x} + x - 1$ , domain: $x \geq 0$ .
$\left(\frac{f}{g}\right)(x) = \frac{\sqrt{x}}{x-1}$ , domain: $x \geq 0$ and $x \neq 1$ .

Composition: Feeding One Machine into Another

This is the most powerful operation. Instead of adding outputs side by side, you take the output of one function and feed it as the input to the other.

(f \circ g)(x) = f\big(g(x)\big)

Read " $f$ composed with $g$ ". You do $g$ first, then $f$ on the result.

Intuition. Suppose $g$ is a machine that converts Celsius to Fahrenheit, and $f$ is a machine that converts Fahrenheit to Kelvin. Then $f \circ g$ converts Celsius directly to Kelvin — one combined machine.

Domain trap. For $f(g(x))$ to make sense, two conditions must hold:

$x$ must be in the domain of $g$ (so $g(x)$ exists).
$g(x)$ must be in the domain of $f$ (so $f$ can accept it).

So the domain of $f \circ g$ is: all $x$ in the domain of $g$ such that $g(x)$ is in the domain of $f$ .

Watch out

Composition is not commutative. $f \circ g$ is almost never the same as $g \circ f$ . For example, if $f(x) = x^2$ and $g(x) = x+1$ , then:

$(f \circ g)(x) = (x+1)^2 = x^2 + 2x + 1$
$(g \circ f)(x) = x^2 + 1$ …

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When combining functions through arithmetic operations, we apply the operation pointwise: $(f \pm g)(x) = f(x) \pm g(x)$ , $(fg)(x) = f(x) \cdot g(x)$ , and $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$ wherever $g(x) \neq 0$ . For $f(x) = 2x + 1$ and $g(x) = x^2 + 1$ , we get four new functions by combining these expressions algebraically.

Understanding Function Operations

When we have two functions, we can create new functions by performing arithmetic operations on their outputs. The key insight is that these operations happen pointwise: for any input $x$ , we first evaluate both functions at that point, then combine the results.

Think of it this way: if $f$ tells you one quantity and $g$ tells you another (both depending on $x$ ), then $f + g$ tells you their sum, $f - g$ their difference, and so on. The domain of the resulting function is typically the intersection of the original domains, with the extra restriction for division that we cannot divide by zero.

Given $f(x) = 2x + 1$ and $g(x) = x^2 + 1$ , both functions are defined for all real numbers, so our combined functions will also be real-valued (except where division by zero might occur).

Finding Each Combined Function

1. Sum: $(f + g)(x)$

By definition, $(f + g)(x) = f(x) + g(x)$ . We substitute the given expressions:

(f + g)(x) = (2x + 1) + (x^2 + 1) = x^2 + 2x + 2

The domain is all real numbers, $\mathbb{R}$ .

2. Difference: $(f - g)(x)$

Similarly, $(f - g)(x) = f(x) - g(x)$ :

(f - g)(x) = (2x + 1) - (x^2 + 1) = 2x + 1 - x^2 - 1 = -x^2 + 2x

Again, the domain is $\mathbb{R}$ .

3. Product: $(fg)(x)$

For the product, $(fg)(x) = f(x) \cdot g(x)$ . We multiply the two expressions:

(fg)(x) = (2x + 1)(x^2 + 1)

Expanding using the distributive property:

= 2x \cdot x^2 + 2x \cdot 1 + 1 \cdot x^2 + 1 \cdot 1 = 2x^3 + 2x + x^2 + 1

Rearranging in standard form:

(fg)(x) = 2x^3 + x^2 + 2x + 1

Domain: $\mathbb{R}$ .

4. Quotient: $\left(\dfrac{f}{g}\right)(x)$ …

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Understanding Function Operations

Finding Each Combined Function

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