Domain and range Knowledge-Based Study

Study notes

1. Fundamental Definitions

   In mathematics, a relation maps elements from an input set to an output set. When each valid input corresponds to exactly one output, this relation is defined as a function.

   For any function $f: A o B$:

   • Domain: The set of all possible input values (independent variable, typically $x$) for which the function is defined and produces a real output.
   • Codomain: The complete set $B$ into which all outputs are constrained to fall.
   • Range: The actual set of all resulting output values (dependent variable, typically $y$ or $f(x)$) produced by substituting every value from the domain into the function. Thus, the range is always a subset of the codomain.

   Mathematically, if $x \in \text{Domain}(f)$, then $f(x) \in \text{Range}(f)$.

   Common Restrictions on Real-Valued Functions

   When working within the set of real numbers $\mathbb{R}$, the natural domain includes all real numbers except those that cause an undefined mathematical operation.

   1. Division by Zero

   A rational expression is undefined whenever its denominator equals zero. For a function in the form $f(x) = \frac{P(x)}{Q(x)}$, the domain requires:

   $$Q(x) \neq 0$$

   2. Even Roots of Negative Numbers

   An even root of a negative number does not yield a real value. For a function in the form $g(x) = \sqrt[2n]{P(x)}$ where $n \in \mathbb{Z}^{+}$, the domain requires:

   $$P(x) \geq 0$$

   3. Combining Restrictions

   If an even root appears in the denominator, such as $h(x) = \frac{1}{\sqrt{P(x)}}$, the radicand must be strictly positive:

   $$P(x) > 0$$

   Approaches to Finding the Range

   Finding the range often requires analyzing the behavior, extrema, or inverse relationships of the function.

   Algebraic Method: Expressing $x$ in Terms of $y$

   For invertible or rational functions, set $y = f(x)$ and solve for $x$ in terms of $y$. Then identify the restrictions on $y$.

   For example, given $f(x) = \frac{2x + 1}{x - 3}$:

   1. Set $y = \frac{2x + 1}{x - 3}$.
   2. Multiply by $x - 3$: $y(x - 3) = 2x + 1 \implies xy - 3y = 2x + 1$.
   3. Group terms containing $x$: $xy - 2x = 3y + 1 \implies x(y - 2) = 3y + 1$.
   4. Solve for $x$: $x = \frac{3y + 1}{y - 2}$.

   Since the denominator cannot be zero, $y \neq 2$. Hence, the range is $(-\infty, 2) \cup (2, \infty)$.

   Graphical Method

   On a Cartesian plane, the domain corresponds to the horizontal spread along the $x$-axis, while the range corresponds to the vertical spread along the $y$-axis from the minimum value to the maximum value.

   Comprehensive Examples

   Example 1: Radical Function

   Find the domain and range of $f(x) = \sqrt{25 - x^2}$.

   • Domain: Set the radicand greater than or equal to zero: $$25 - x^2 \geq 0 \implies x^2 \leq 25 \implies -5 \leq x \leq 5$$ In interval notation, the domain is $[-5, 5]$.
   • Range: Since $x^2 \geq 0$, the radicand $25 - x^2$ takes values from 0 (when $x = \pm 5$) to 25 (when $x = 0$). Taking the principal square root yields values from $\sqrt{0} = 0$ to $\sqrt{25} = 5$. Therefore, the range is $[0, 5]$.

   Example 2: Quadratic Function

   Find the domain and range of $g(x) = x^2 - 4x + 7$.

   • Domain: As a polynomial function, $g(x)$ has no division by zero or radical restrictions. The domain is $(-\infty, \infty)$.
   • Range: Completing the square gives $g(x) = (x - 2)^2 + 3$. Since $(x - 2)^2 \geq 0$ for all real $x$, the minimum value of $g(x)$ is 3, occurring at $x = 2$. The range is $[3, \infty)$.