Functions are at the heart of algebra, providing a systematic way to express the relationship between variables. This section introduces the concept of functions, their domain and range, and how to use function notation, focusing on mathematical precision and clarity.

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**What is a Function?**

A function defines a specific relationship where each input value (x) corresponds to exactly one output value, denoted as (f(x)). Functions can be represented in various forms: as equations, graphs, or tables.

**Domain and Range**

**Domain**

The **domain** of a function is the set of all possible input values $x$ for which the function is defined.

**Range**

The **range** of a function includes all possible output values $(f(x))$ that result from using the domain.

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The domain and range of this function $f(x) = 2x$ is given as domain $D = {x ∈ N}$, range $R = {y ∈ N: y = 2x}$.

**Function Notation**

Function notation, such as $f(x)$, provides a convenient way to reference functions and perform operations with them.

## Worked Examples

**Example 1: Linear Function **$f(x) = 3x - 5$

**Domain**: All real numbers $(\mathbb{R})$.**Range**: All real numbers $(\mathbb{R})$.

**Example 2: Rational Function **$g(x) = \frac{3(x + 4)}{5}$

**Domain**: All real numbers $(\mathbb{R})$.**Range**: All real numbers $(\mathbb{R})$.

**Practical Problems**

**Problem 1: **

**Find the output for **$f(x) = 3x - 5$** when **$x = 3$**.**

**Given: **$f(x) = 3x - 5$

**Solution:**

To find $f(3)$:

$f(3) = 3(3) - 5 = 9 - 5 = 4$**Problem 2: **

**Find the output for **$g(x) = \frac{3(x + 4)}{5}$** when **$x = 1$**.**

Given: $g(x) = \frac{3(x + 4)}{5}$

**Solution:**

To find $g(1)$:

$g(1) = \frac{3(1 + 4)}{5} = \frac{3(5)}{5} = \frac{15}{5} = 3$