Function Mappings and Properties for the TMUA

The definition of a function

In simple terms, a function is a rule, or mapping, that connects a set of input values to a set of output values. However, not every algebraic expression or mapping qualifies as a function. To define a function properly, three specific criteria must be met:

1. There must be a rule, usually an algebraic expression, that maps input values to output values.
2. There must be a clear list of permitted input values, which is known as the Domain of the function.
3. For every single input value in the domain, the rule must produce exactly one output value.

To illustrate this, consider the expression $f(x) = x^2$. This is a function because every number you square results in a single, unique value. In contrast, the expression $f(x) = \pm\sqrt{x}$ is not a function because a single input, such as $x = 16$, leads to two different output values, $4$ and $-4$. The only exception to this in the $\pm\sqrt{x}$ example is when $x = 0$, but a function must provide a single output for every value in its domain.

Domain and notation

While a function is strictly denoted by a letter like $f$ and its value as $f(x)$, it is common in the TMUA to see the notation simplified to $y = f(x)$. The Domain is the set of all $x$ values for which the function is defined. Usually, the domain is assumed to be the set of all real numbers unless stated otherwise. Some functions have inherent restrictions: for instance, the square root function is only defined for $x \ge 0$, and logarithmic functions are only defined for positive $x$ values. Occasionally, a domain is restricted manually for a specific problem, such as defining $f(x) = x^2 + 3$ for $x \ge 2$.

One-to-one and many-to-one mappings

While every input must have only one output, it is possible for multiple different inputs to result in the same output. This leads to two categories of functions:

• One-to-one functions: Each unique input produces a unique output, and no two inputs share an output. For example, $g(x) = x^3$ is one-to-one because every real number has a unique cube.
• Many-to-one functions: Different input values can lead to the same output value. For example, $f(x) = x^2$ is many-to-one because both $f(2) = 4$ and $f(-2) = 4$.

Graphical tests for functions

You can identify a function and its mapping type by inspecting its graph using two simple geometric tests:

1. The Vertical Line Test: This test determines if a mapping is a function. If any vertical line drawn on the graph crosses the curve more than once, it is not a function because that $x$ value has multiple $y$ values.

1. The Horizontal Line Test: Once you know a graph represents a function, this test determines the mapping type. if every horizontal line crosses the function at most once, it is a one-to-one function.

If a horizontal line can be drawn that crosses the function in more than one place, then the function is many-to-one.

Properties of common functions

There are two common functions whose properties you must know by heart for the TMUA:

The Square Root Function $f(x) = \sqrt{x}$

By mathematical convention, the symbol $\sqrt{a}$ always refers to the positive square root. This convention is adopted without comment in TMUA questions: $\sqrt{64}$ is 8, and not $-8$. The domain is restricted to $x \ge 0$. As seen in its graph, it is a one-to-one function.

The Modulus Function $f(x) = |x|$

The modulus function, or absolute value, takes the positive value of whatever is inside the vertical lines. Thus, $|7| = 7$ and $|-2| = 2$. To sketch the graph of $y = |f(x)|$, you first sketch the original function $y = f(x)$ and then reflect any part of the graph that lies below the $x$-axis in the $x$-axis, while leaving the parts above the $x$-axis unchanged.