IB AA: Functions

This chapter introduces the formal concept of a function as a rule that assigns each input value to exactly one output value. You will learn to work with the core components of a function: its domain (the set of all possible inputs) and its range (the set of all possible outputs). The chapter covers identifying and graphing different families of functions, including linear, quadratic, exponential, and rational functions. A key skill is using the vertical line test to determine if a graph represents a function.

The topic then moves to combining and manipulating functions. You will learn how to create composite functions, where the output of one function becomes the input for another. You will also study inverse functions, which reverse the action of the original function. This involves finding the equation of an inverse, understanding its graphical relationship to the original function (a reflection in the line y=x), and knowing the conditions under which an inverse exists. The abstract nature of composite and inverse functions, particularly determining their domains, is often where students find the material most challenging. This chapter builds heavily on prior knowledge of algebraic manipulation and graphing.

Formulas for IB AA: Functions

Composite function: $(f \circ g)(x) = f(g(x))$ The output of function g becomes the input for function f.
Inverse function composition: $(f^{-1} \circ f)(x) = x$ Composing a function with its inverse returns the original input, x.
Vertical translation: $y = f(x) + k$ The graph of f(x) is shifted up by k units for k > 0, and down for k < 0.
Horizontal translation: $y = f(x - h)$ The graph of f(x) is shifted right by h units for h > 0, and left for h < 0.
Vertical stretch or compression: $y = a f(x)$ Stretches the graph of f(x) vertically by a factor of |a|. If a < 0, it is also reflected in the x-axis.
Horizontal stretch or compression: $y = f(bx)$ Stretches the graph of f(x) horizontally by a factor of 1/|b|. If b < 0, it is also reflected in the y-axis.
Reflection in the x-axis: $y = -f(x)$ Reflects the graph of f(x) across the x-axis.
Reflection in the y-axis: $y = f(-x)$ Reflects the graph of f(x) across the y-axis.