Q.Express the following functions as set of ordered pairs and determine their range. $f : X \to \mathbf{R}$, $f(x) = x^3 + 1$, where $X = \{-1, 0, 3, 9, 7\}$

Concept: Function as a set of ordered pairs — each input $x$ from the domain $X$ is paired with exactly one output $f(x)$.

Step 1: Compute $f(x) = x^3 + 1$ for each $x \in X$:

• $f(-1) = (-1)^3 + 1 = -1 + 1 = 0$
• $f(0) = 0^3 + 1 = 1$
• $f(3) = 27 + 1 = 28$
• $f(9) = 729 + 1 = 730$
• $f(7) = 343 + 1 = 344$

Step 2: Write the set of ordered pairs:

$\{(-1, 0), (0, 1), (3, 28), (9, 730), (7, 344)\}$ …

The core idea here is simple: a function is a rule that pairs each input with exactly one output. When the domain is a finite set like $X$, we can list every pair explicitly. The range is just the collection of all outputs that actually appear.

Think of an arrow diagram: draw the set $X$ on the left, $\mathbf{R}$ on the right, and for each $x$ in $X$, draw an arrow to $f(x)$. The ordered pairs are just the coordinates of those arrows. The range is the set of all right-hand endpoints that get hit.

Let’s work through each element of $X$ one by one.

1. For $x = -1$:

   $f(-1) = (-1)^3 + 1 = -1 + 1 = 0$.

   So the ordered pair is $(-1, 0)$.

2. For $x = 0$:

   $f(0) = 0^3 + 1 = 0 + 1 = 1$.

   Ordered pair: $(0, 1)$.

3. For $x = 3$:

   $f(3) = 3^3 + 1 = 27 + 1 = 28$.

   Ordered pair: $(3, 28)$.

4. For $x = 9$:

   $f(9) = 9^3 + 1 = 729 + 1 = 730$.

   Ordered pair: $(9, 730)$.

5. For $x = 7$:

   $f(7) = 7^3 + 1 = 343 + 1 = 344$.

   Ordered pair: $(7, 344)$.

Now collect all these pairs into a set. The order of listing doesn’t matter, but it’s tidy to keep them in the same order as the domain. …