Mathematical Functions, Domains, and Ranges

Domain and Range

Every function is defined by two critical sets of values: the Domain and the Range.

• Domain ($D$): The set of all possible input values (usually $x$) for which the function is defined.
• Range ($R$): The set of all resulting output values (usually $y$) that the function produces.

When working with square roots or fractions, we must be careful to identify restrictions. For instance, the square root of a negative number is not defined in the set of real numbers. Consider the function:

$$y = \sqrt{x-2}$$

To find the domain, we must ensure the expression inside the radical is non-negative:

$$x - 2 \geq 0 \implies x \geq 2$$

Thus, the domain is $D = \{x \in \mathbb{R} \mid x \geq 2\}$, or in interval notation, $[2, \infty)$. The resulting range for this specific function is $R = \{y \in \mathbb{R} \mid y \geq 0\}$, or $[0, \infty)$.

If we change the function to $y = \frac{1}{\sqrt{x-2}}$, the domain changes because the denominator cannot be zero. In this case, $x-2$ must be strictly greater than zero, so $D = (2, \infty)$.

Exam Tip: When finding the domain of a rational function (a fraction), always set the denominator to not equal zero. When finding the domain of a square root, always set the radicand (the part inside) to be greater than or equal to zero.

Function Evaluation

Function evaluation is the process of determining the output value of a function for a specific input. We often use the notation $f(x)$, where $x$ is the input.

If we have a function $f(\xi) = \xi^2 + 2\xi - 1$, and we want to evaluate it at $\xi = -1$, we substitute $-1$ everywhere we see $\xi$:

$$\begin{aligned} f(-1) &= (-1)^2 + 2(-1) - 1 \\ &= 1 - 2 - 1 \\ &= -2 \end{aligned}$$

We can also evaluate functions using algebraic expressions instead of just single numbers. For example, if $g(\xi) = \xi^2$, then evaluating $g(3x+4)$ means replacing the $\xi$ with the entire expression $(3x+4)$:

$$\begin{aligned} g(3x+4) &= (3x+4)^2 \\ &= 9x^2 + 16 + 2(3x)(4) \\ &= 9x^2 + 24x + 16 \end{aligned}$$

Piecewise Functions

Sometimes, a single formula isn't enough to describe a relationship. A piecewise function is a function that uses different formulas for different parts of its domain.

Consider the following example:

$$f(x) = \begin{cases} x + 3 & \text{if } -1 \leq x \leq 2 \\ x^2 & \text{if } x > 2 \end{cases}$$

In this function, the domain is $[-1, \infty)$. To evaluate the function, you must first check which "piece" the input falls into.

• To find $f(1)$: Since $1$ is between $-1$ and $2$, we use the first rule: $1 + 3 = 4$.
• To find $f(3)$: Since $3$ is greater than $2$, we use the second rule: $3^2 = 9$.

Function Composition

Composition occurs when one function is applied to the result of another. This is often written as $(f \circ g)(x)$, which is equivalent to $f(g(x))$.

Let's look at an example with two functions:

$$f(x) = x^2 + 1$$ $$g(x) = 3x + 2$$

To find the composition $f(g(x))$, we plug the entire $g(x)$ function into $f(x)$:

$$\begin{aligned} f(g(x)) &= (3x + 2)^2 + 1 \\ &= (9x^2 + 12x + 4) + 1 \\ &= 9x^2 + 12x + 5 \end{aligned}$$

Conversely, to find $g(f(x))$, we plug $f(x)$ into $g(x)$:

$$\begin{aligned} g(f(x)) &= 3(x^2 + 1) + 2 \\ &= 3x^2 + 3 + 2 \\ &= 3x^2 + 5 \end{aligned}$$

Note that $f(g(x))$ is generally not equal to $g(f(x))$. The order of composition matters significantly.