Introduction

In mathematics, a function is a relationship between a set of inputs (called the domain) and a set of outputs (called the range) where each input is related to exactly one output. Functions play a crucial role in various mathematical concepts and real-life applications.

Definition of a Function

A function $f$ from set $A$ to set $B$ is a rule or relation that assigns to each element $x$ in set $A$ exactly one element $y$ in set $B$. This is denoted as $f: A \rightarrow B$.

Example:

Consider a function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = 2x + 3$. Find $f(4)$.

• Substitute $x = 4$ into the function: $f(4) = 2(4) + 3 = 8 + 3 = 11$. Therefore, $f(4) = 11$.

Domain and Range of a Function

• Domain: The set of all possible inputs for a function.
• Range: The set of all possible outputs for a function.

Example:

Given the function $g: \mathbb{R} \rightarrow \mathbb{R}$ defined by $g(x) = x^2 - 1$. Find the domain and range of $g$.

• Domain: Since $x^2 - 1$ is defined for all real numbers, the domain of $g$ is $\mathbb{R}$.
• Range: To find the range, consider the values that $g(x)$ can take. Since $x^2 \geq 0$ for all $x$, $g(x) = x^2 - 1 \geq -1$. So, the range of $g$ is $[-1, \infty)$.

Types of Functions

1. Linear Function: A function of the form $f(x) = mx + c$, where $m$ and $c$ are constants.
2. Quadratic Function: A function of the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.
3. Exponential Function: A function of the form $f(x) = a \cdot b^x$, where $a$ and $b$ are constants and $b > 0$.
4. Logarithmic Function: A function of the form $f(x) = \log_b(x)$, where $b$ is the base of the logarithm.

Example:

Consider the function $h(x) = 2x^2 - 3x + 1$. Determine the type of function represented by $h(x)$.

• Since $h(x)$ is a quadratic function of the form $ax^2 + bx + c$, it is a quadratic function.

Composite Functions

A composite function is formed by applying one function to the result of another function.

Let $f: A \rightarrow B$ and $g: B \rightarrow C$ be two functions. Then, the composite function $g \circ f$ is defined as $(g \circ f)(x) = g(f(x))$.

Example:

Given $f(x) = 2x + 1$ and $g(x) = x^2$, find the composite function $g \circ f$.

• First, find $f(x)$ and then substitute it into $g$: $(g \circ f)(x) = g(f(x)) = g(2x + 1) = (2x + 1)^2$.

Inverse Functions

An inverse function undoes the effect of another function. If $f$ is a function from set $A$ to set $B$, the inverse function of $f$, denoted as $f^{-1}$, is such that $f^{-1}(f(x)) = x$ for all $x$ in $A$.

Example:

Consider the function $p(x) = 3x - 4$. Find the inverse of $p(x)$.

• Let $y = 3x - 4$. To find the inverse, swap $x$ and $y$: $x = 3y - 4$. Solve for $y$ to get $y = \frac{x + 4}{3}$. Therefore, the inverse of $p(x)$ is $p^{-1}(x) = \frac{x + 4}{3}$.

Common Mistakes

• Confusing domain and range: Ensure to correctly identify the domain and range of a function.
• Misunderstanding composite functions: Be careful when applying one function to the output of another function in composite functions.

Key Points

• Functions relate inputs to outputs in a unique manner.
• The domain is the set of all possible inputs, and the range is the set of all possible outputs.
• Different types of functions include linear, quadratic, exponential, and logarithmic functions.
• Composite functions combine two functions by applying one to the result of the other.
• Inverse functions undo the effect of another function.

Practice Questions

1. Let $f(x) = 4x - 2$ and $g(x) = x^2$. Find $(f \circ g)(x)$.

Answer: First, find $g(x)$ and then substitute it into $f$: $(f \circ g)(x) = f(g(x)) = f(x^2) = 4x^2 - 2$.

2. Consider the function $m(x) = 5x^3 + 2x$. Determine the domain and range of $m$.

Answer:

• Domain: Since $m(x)$ is defined for all real numbers, the domain is $\mathbb{R}$.
• Range: To find the range, consider the behavior of cubic functions. As $x^3$ can take any real value, the range of $m$ is also $\mathbb{R}$.

3. Given $h(x) = \frac{2x - 1}{x + 3}$, determine the inverse function of $h(x)$.

Answer: Let $y = \frac{2x - 1}{x + 3}$. Swap $x$ and $y$, and solve for $y$ to find the inverse function $h^{-1}(x)$.

4. If $q(x) = 3^x$, identify the type of function represented by $q(x)$.

Answer: The function $q(x)$ is an exponential function of the form $a \cdot b^x$, where $a = 1$ and $b = 3$.

5. Find the composite function $f \circ f$ if $f(x) = x^2 - 3x + 2$.

Answer: To find $f \circ f$, compute $f(f(x))$ by substituting $f(x)$ into itself and simplifying the expression.