Functions

Introduction

In mathematics, a function is a relationship between a set of inputs (called the domain) and a set of possible outputs (called the range), where each input is related to exactly one output. Functions are widely used in various mathematical and real-world applications to model relationships between variables.

Definition of a Function

A function $f: A \rightarrow B$ is a rule or correspondence that assigns to each element $x$ in the set $A$ exactly one element, called $f(x)$, in the set $B$. Here, $A$ is the domain of the function and $B$ is the codomain.

Example 1:

Define a function $f: \mathbb{R} \rightarrow \mathbb{R}$ by $f(x) = 2x + 3$.

For $x = 4$, we have:

$$f(4) = 2(4) + 3 = 11$$

Therefore, $f(4) = 11$.

Types of Functions

1. Linear Function:

A function of the form $f(x) = mx + c$, where $m$ and $c$ are constants, is called a linear function.

Example 2:

Define a linear function $g(x) = 3x - 2$.

For $x = 5$, we have:

$$g(5) = 3(5) - 2 = 13$$

Therefore, $g(5) = 13$.

2. Quadratic Function:

A function of the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants and $a \neq 0$, is called a quadratic function.

Example 3:

Define a quadratic function $h(x) = 2x^2 + 5x - 3$.

For $x = -2$, we have:

$$h(-2) = 2(-2)^2 + 5(-2) - 3 = 1$$

Therefore, $h(-2) = 1$.

3. Exponential Function:

A function of the form $f(x) = a \cdot b^x$, where $a$ and $b$ are constants and $b > 0$, $b \neq 1$, is called an exponential function.

Example 4:

Define an exponential function $f(x) = 2 \cdot 3^x$.

For $x = 0$, we have:

$$f(0) = 2 \cdot 3^0 = 2$$

Therefore, $f(0) = 2$.

Common Mistakes

1. Confusing Functions with Relations: Remember that a function is a special type of relation where each input has exactly one output. Not all relations are functions.

2. Misinterpreting Domain and Range: Be careful to correctly identify the domain and range of a function. The domain is the set of all possible inputs, while the range is the set of all possible outputs.

Key Points

• A function maps each element from its domain to a unique element in its range.
• Types of functions include linear, quadratic, exponential, and more.
• Domain and range are important concepts when studying functions.

Practice Questions

1. Define a linear function $f(x) = 4x - 7$. Calculate $f(3)$.

Answer: For $x = 3$, $$f(3) = 4(3) - 7 = 5$$ Therefore, $f(3) = 5$.

2. Consider the quadratic function $g(x) = x^2 - 6x + 9$. Find $g(3)$.

Answer: For $x = 3$, $$g(3) = (3)^2 - 6(3) + 9 = 0$$ Therefore, $g(3) = 0$.

3. Determine whether the following relation is a function: ${(1, 2), (3, 4), (1, 5)}$.

Answer: This relation is not a function because the input $1$ is associated with two different outputs, $2$ and $5.

4. Define an exponential function $h(x) = 3 \cdot 2^x$. Calculate $h(-1)$.

Answer: For $x = -1$, $$h(-1) = 3 \cdot 2^{-1} = \frac{3}{2}$$ Therefore, $h(-1) = \frac{3}{2}$.

5. Is the function $f(x) = x^2 + 1$ linear? Explain your answer.

Answer: No, the function $f(x) = x^2 + 1$ is not linear because it contains a squared term, making it a quadratic function.