Introduction to Domain and Range

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Fundamental Definitions: Domain and Range

• Domain

  • The domain represents the set of all possible input values (typically $x$) that can be plugged into a function.

• Range

  • The range consists of the set of all output values (typically $y$) generated by the function.

Analysis of Domain and Range from Graphs

• Example 1: Continuous Interval

  • Based on a specific graph, the input values extend along the $x$-axis from negative five to five.

  • The domain is identified as the interval $[-5, 5)$, where the square bracket indicates inclusion and the parenthesis indicates exclusion.

  • The corresponding range, found by observing the vertical extent of the graph, is the interval $[-2, 5)$.

• Example 2: Parabola with Infinite Extent

  • Arrows at the top of a graph indicate that the function continues in that direction forever.

  • In the case of a standard parabola opening upwards with its lowest point at $(1, -4)$, any real number can be used as an input.

  • Domain: $(-\text{infinity}, \text{infinity})$ or $(-\infty, \infty)$.

  • Range: The lowest value is $-4$ (inclusive). The graph grows infinitely, so the range is $[-4, \infty)$.

  • Note: It is permissible for multiple $x$ values to produce the same $y$ value (e.g., plugging in $3$ and $-1$ both result in an output of $4$); this does not alter the range calculation.

• Example 3: Discontinuous Segments

  • When a graph is composed of disconnected segments or contains holes (open circles), the domain and range must be written as unions of intervals.

  • In a graph with open circles at $x = -3$, $x = 0$, and $x = 5$, but a closed circle at $x = 1$, the domain is defined as $(-3, 0) \cup [1, 5)$.

  • The range for this specific graph is defined from its lowest point to its highest. Despite an open circle at one terminal, if that $y$ value is reached elsewhere on the graph, it is included. For instance, if an open circle exists at $(2, -4)$ but the point $(2, -4)$ is actually defined elsewhere, or the graph reaches $-5$ at a closed circle, the interval reflects the actual outputs attained.

  • Example range: $[-5, 2)$.

• Example 4: Fragmented Chunks

  • A graph split into two distinct horizontal chunks may have a domain such as $(-4, -1] \cup (1, 3]$.

  • If the lowest output reached is $-3$ (exclusive) and the highest is $5$ (inclusive), the range is $(-3, 5]$, even if there is an open circle at a specific height (like $y = 0$) elsewhere on the page, provided that height is reached by a different part of the function.

Mathematical Properties of Polynomial Functions

• Defining a Polynomial

  • A polynomial is a function consisting of the sum of multiple terms, where each term is a coefficient multiplied by $x$ raised to a non-negative whole number power.

  • Examples of polynomial terms include $x^3$, $2x$ (which is $2x^1$), and $1$ (which is $1x^0$).

• Domain of Polynomials

  • There is no real number that, when substituted into a polynomial, causes the math to "break." You can multiply any number by a constant and add other constants without restriction.

  • Every polynomial has a domain of $(-\infty, \infty)$.

Rational Functions and Excluded Values

• The Division by Zero Constraint

  • Mathematical errors arise when a denominator becomes zero. To find the domain of a rational function, one must identify and exclude values that result in a zero denominator.

• Example 1: $f(x) = \frac{5}{x+3}$

  • Strategy: Set the denominator to zero: $x + 3 = 0$.

  • Solution: $x = -3$.

  • Conclusion: The domain is everything except $-3$. Written as an interval: $(-\infty, -3) \cup (-3, \infty)$.

• Example 2: $f(x) = \frac{x+2}{2x-7}$

  • The numerator being zero is acceptable (the function value simply becomes zero). The concern is the denominator.

  • Strategy: Set $2x - 7 = 0$.

  • Steps: $2x = 7 \Rightarrow x = \frac{7}{2}$.

  • Conclusion: The domain is everything except $\frac{7}{2}$. Written as an interval: $(-\infty, \frac{7}{2}) \cup (\frac{7}{2}, \infty)$.

Domain Requirements for Radical Functions

• Square Roots and Real Numbers

  • In the real number system, the radicand (the expression inside the square root) cannot be negative. While the square root of $-1$ is the imaginary unit $i$, for real-valued functions, the radicand must be greater than or equal to zero.

• Example: $f(x) = \sqrt{-3x + 9}$

  • Strategy: Set the radicand to be $\geq 0$.

  • Inequality: $-3x + 9 \geq 0$.

  • Steps: $-3x \geq -9$.

  • Rule: When multiplying or dividing an inequality by a negative number, the inequality sign must be flipped.

  • Result: $x \leq \frac{-9}{-3} \Rightarrow x \leq 3$.

  • Domain: $(-\infty, 3]$.

• Combined Constraints (Square Root in Denominator)

  • Example: $f(x) = \frac{5}{\sqrt{x}}$.

  • Constraint 1: The denominator cannot be zero ($\sqrt{x} \neq 0 \Rightarrow x \neq 0$).

  • Constraint 2: The radicand cannot be negative ($x \geq 0$).

  • Combined Strategy: The value of $x$ must be strictly greater than zero.

  • Domain: $(0, \infty)$.

Behavior of Even vs. Odd Roots

• Odd Roots (e.g., Cube Roots)

  • The cube root ($\sqrt[3]{x}$) behaves differently than the square root. One can take the odd root of a negative number.

  • Proof through expansion:

    • $(-1)^3 = (-1) \times (-1) \times (-1) = -1$, therefore $\sqrt[3]{-1} = -1$.

    • $(-2)^3 = (-2) \times (-2) \times (-2) = -8$, therefore $\sqrt[3]{-8} = -2$.

  • Because odd roots of negative numbers are real, there are no domain restrictions based on the root itself.

  • A function like $f(x) = \sqrt[3]{5}$ (or any cube root of a polynomial) has a domain of $(-\infty, \infty)$.

Questions & Discussion

• Question: Can we remind the class when to flip the sign in an inequality?

• Response: The sign must be flipped only when dividing or multiplying by a negative number.

• Question: What makes the function $f(x) = \frac{5}{\sqrt{x}}$ weirder than previous examples?

• Response: It contains both a square root and a denominator. We must satisfy two conditions: the denominator must be non-zero and there can be no negatives inside the square root.

• Question: What is the singular form of the word "parentheses" and "brackets"?

• Response: The singular of "brackets" is "bracket." The singular of "parentheses" is "parenthesis."