Additional Mathematics

Core

# Functions

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Functions

**Function Definitions and Properties**

- A
**function**is a relationship between input and output values, where each input corresponds to exactly one output. - The set of all input values is known as the
**domain**, while the set of output values is the**range**. - The input is often represented by the variable
**x**, and the output by the variable**y**.

**Types of Functions**

- A
**linear function**has a constant rate of change. It is represented by the formula**y = mx + c**, where**m**is the slope, and**c**the y-intercept. - A
**quadratic function**is defined by a polynomial of second degree. It is represented by the formula**y = ax² + bx + c**. - An
**exponential function**is a function in which the variable appears in the exponent, such as**y = a^x**. - The
**logarithmic function**is the inverse of the exponential function and has the form**y = loga x**.

**Composition of Functions**

- When two functions are combined in such a way that the output of one becomes the input of the other, this is known as the
**composition**of functions. - If
**f**and**g**are functions, the composition of**f**and**g**is written as**f(g(x))**or**(f o g)(x)**, read as 'f of g of x'.

**Inverse of a Function**

- The
**inverse**of a function, denoted by**f^-1(x)**, is a function that 'undoes' the action of the original function. - If a function
**f**takes a value**x**and transforms it into**y**, its inverse function**f^-1**takes the value**y**and transforms it back to**x**.

**Graphs of Functions**

- The graph of a function is a visual representation of the relationship between input and output values.
- All points (x, y) on the graph represent valid input-output pairs, where the
**x**-coordinate corresponds to the input value and the**y**-coordinate to the output value. **Direction**of graphs can be determined by viewing the left-to-right progression of a graph. If values of y increase as x increases, the function is said to be**increasing**. If the values of y decrease as x increases, the function is said to be**decreasing**.

**Transformations of Functions**

- A function can be transformed by changing the form of its rule or equation.
- Common transformations include
**translations**(shifting the whole graph left, right, up, or down),**stretches and compressions**(altering the shape of the graph without changing its location), and**reflections**(flipping the graph about a line).