Range in inequalities refers to the set of all possible output values, often denoted as y or f(x), that a function can produce when its input values (domain) are restricted or defined by inequalities. It represents the vertical extent of a function's graph.
Understanding Range: The Output Values
At its core, range is the set of all y values, the dependent quantity, that will result from substituting all x values (the domain) into the function. In simpler terms, if you consider a mathematical relationship or function, the range comprises every possible output you can get from that relationship. The output values depend on the input values and the rule connecting them.
- Dependent Quantity (y or f(x)): These are the output values, which form the range.
- Independent Quantity (x): These are the input values, which form the domain.
How Inequalities Influence Range
Inequalities play a crucial role in determining the range of a function, primarily by defining or restricting its domain. When the allowed input values (the domain) are limited by an inequality, the set of possible output values (the range) is consequently constrained.
- Defining the Domain: Often, an inequality is used to specify which input values (x) are valid for a function. For instance, in the function $f(x) = \sqrt{x - 2}$, the expression under the square root cannot be negative. This imposes the inequality $x - 2 \ge 0$, meaning $x \ge 2$. This domain restriction directly impacts the minimum possible value of $f(x)$, thus defining the range.
- Constraining Function Behavior: Some functions, like piecewise functions, are defined by different rules over different parts of their domain, with these parts separated by inequalities. Each piece contributes to the overall range.
- Representing Regions (Solution Sets): When an entire inequality describes a relationship (e.g., $y > 2x + 1$), it represents a region in the coordinate plane rather than the range of a single function $y = f(x)$. In such cases, the "range" refers to the set of y-coordinates of all points that satisfy the inequality. This is often described as the solution set or region of solutions.
Practical Examples of Range with Inequalities
Let's explore how inequalities affect the range in various functional contexts:
-
Linear Function with a Restricted Domain:
- Function: $f(x) = 2x + 3$
- Domain (defined by an inequality): $x > 1$
- To find the range, substitute the boundary condition into the function. If $x > 1$, then $2x > 2$, and $2x + 3 > 2 + 3$, so $f(x) > 5$.
- Range: $(5, \infty)$
-
Quadratic Function with No Explicit Domain Restriction:
- Function: $f(x) = x^2 - 4$
- Domain (all real numbers): $(-\infty, \infty)$
- Since $x^2 \ge 0$ for all real $x$, then $x^2 - 4 \ge -4$. The minimum value of $f(x)$ is $-4$.
- Range: $[-4, \infty)$
-
Square Root Function (Implied Domain by Inequality):
- Function: $f(x) = \sqrt{x - 5}$
- Domain (implied by inequality): For $\sqrt{x-5}$ to be a real number, $x - 5 \ge 0$, so $x \ge 5$.
- Since the square root symbol denotes the principal (non-negative) square root, $\sqrt{x - 5} \ge 0$.
- Range: $[0, \infty)$
-
Rational Function (Implied Domain by Inequality):
- Function: $f(x) = \frac{1}{x + 2}$
- Domain (implied by inequality): The denominator cannot be zero, so $x + 2 \ne 0$, meaning $x \ne -2$.
- As $x$ approaches $-2$ from either side, $|f(x)|$ approaches infinity. As $|x|$ approaches infinity, $f(x)$ approaches $0$. Since $f(x)$ can be positive or negative but never exactly $0$,
- Range: $(-\infty, 0) \cup (0, \infty)$
Here's a summary table:
| Function | Domain (Input values) | Range (Output values) |
|---|---|---|
| $f(x) = 2x + 3$ for $x > 1$ | $(1, \infty)$ | $(5, \infty)$ |
| $f(x) = x^2 - 4$ | $(-\infty, \infty)$ | $[-4, \infty)$ |
| $f(x) = \sqrt{x - 5}$ | $[5, \infty)$ | $[0, \infty)$ |
| $f(x) = \frac{1}{x + 2}$ | $(-\infty, -2) \cup (-2, \infty)$ | $(-\infty, 0) \cup (0, \infty)$ |
Expressing Range: Interval Notation
Range is typically expressed using interval notation, which is a concise way to represent sets of real numbers.
- Parentheses ( ): Indicate that the endpoint is not included (for strict inequalities like $<$, $>$ or for infinity).
- Brackets [ ]: Indicate that the endpoint is included (for non-strict inequalities like $\le$, $\ge$).
- Infinity ($\infty$ or $-\infty$): Always used with parentheses, as infinity is not a number that can be included.
- Union ($\cup$): Used to combine multiple intervals if the range is discontinuous.
Key Takeaways
- The range defines all possible output values ($y$ or $f(x)$) of a function.
- Inequalities often determine the domain (allowed input values) of a function, which in turn directly shapes its range.
- Understanding domain and range is fundamental for analyzing function behavior and graphing.
- Range is distinct from a solution set of an inequality that describes a region, although both involve sets of values.
