Functions

Understanding Functions

• A function is a special type of relation in which every input (also known as an element of the domain) is associated with exactly one output (an element of the range).
• When describing functions, the notation f(x) = y is often used. Here f is the function, x is the input value, and y is the output value.
• The set of all possible input values is called the domain. The set of all resulting output values is called the range.

Types of Functions

• Linear functions take the form y = mx + c, where m is the slope and c is the y-intercept. The graph of a linear function is always a straight line.
• Quadratic functions have the form y = ax^2 + bx + c, where a ≠ 0. The graph of a quadratic function is a parabola.
• Exponential functions are of the form f(x) = a • b^x, where a ≠ 0, b > 0, and b ≠ 1. The graph of an exponential function shows exponential growth if b > 1 and exponential decay if 0 < b < 1.
• Logarithmic functions are the inverse of exponential functions and have the form f(x) = log_b(x). They are used to solve equations where the variable is in the exponent.
• Trigonometric functions, including sine, cosine, and tangent, are used extensively in modelling periodic behaviour and in geometry. The input for these functions is an angle.

Composite Functions

• Composite functions (denoted f(g(x)) or (f o g)(x)) consist of one function nested inside another. To evaluate a composite function, first apply the inner function to the input, and then apply the outer function.
• Composite functions allow for more complex behaviour than can be achieved by a single function alone.

Inverse Functions

• An inverse function exists if a function’s input can be calculated from its output. It is denoted as f^-1.
• The graph of a function and its inverse are mirror images across the line y = x.
• To find the inverse function, swap the roles of y and x, and then solve for y.

Transformations of Functions

• Functions can be transformed by stretching or compressing (scaling), translating (shifting), or reflecting them.
• For a function f, the graph of af(x) is a vertical stretch by factor a, f(ax) is a horizontal compression by a, f(x + b) is a shift to the left by b units, f(x) + b is a shift upwards by b units.
• The graph of -f(x) is a reflection in the x-axis and the graph of f(-x) is a reflection in the y-axis.

Function Limitations

• Not all rules for calculating output from an input constitute a function. Rules which associate an input with more than one output are not functions. These can be identified by the vertical line test, where any vertical line that intersects the graph at more than one point means it is not a function.
• Functions may also have undefined points, sometimes caused when a function is defined by an equation that requires dividing by zero or taking the square root of a negative number. These points are known as discontinuities.