Functions: Domain and Range – Precalculus Study Notes

Functions and Graphs

Domain and Range of Functions

Understanding the domain and range of a function is fundamental in precalculus. The domain refers to all possible input values (typically x-values) for which the function is defined, while the range refers to all possible output values (typically y-values) that the function can produce.

• Domain: The set of all real numbers x for which the function f(x) is defined.

• Range: The set of all real numbers y that the function f(x) can output.

• Determining Domain: To find the domain, identify values of x that do not result in undefined expressions (such as division by zero or taking the square root of a negative number).

• Determining Range: To find the range, consider the possible values of f(x) as x varies over the domain.

Example:

• For the function $f(x) = \frac{1}{x-2}$, the domain is all real numbers except $x = 2$, since division by zero is undefined.

• For the function $g(x) = \sqrt{x+3}$, the domain is $x \geq -3$, since the square root is only defined for non-negative arguments.

Application: Determining the domain and range is essential before graphing a function or solving equations involving functions.

Critical Values and Extrema

Critical values are points in the domain of a function where the function reaches a maximum or minimum value, or where its behavior changes significantly.

• Maximum Value: The largest output (y-value) that a function attains on its domain.

• Minimum Value: The smallest output (y-value) that a function attains on its domain.

• Extrema: The collective term for maximum and minimum values.

• Finding Extrema: Extrema can often be found by analyzing the function's graph or by solving $f'(x) = 0$ for differentiable functions (additional info: calculus methods are used for continuous functions).

Example:

• For $f(x) = x^2$, the minimum value occurs at $x = 0$, where $f(0) = 0$.

• For $f(x) = -x^2 + 4$, the maximum value occurs at $x = 0$, where $f(0) = 4$.

Summary Table: Domain and Range Determination

Function Type	Domain	Range	Notes
Polynomial	All real numbers	Depends on degree and leading coefficient	No restrictions
Rational	All real numbers except where denominator = 0	Depends on numerator/denominator	Check for vertical asymptotes
Radical (even root)	Set inside root $\geq 0$	Non-negative outputs	Domain limited by radicand
Exponential	All real numbers	Positive real numbers	Range never zero or negative

Additional info: In calculus, critical points are found using derivatives, but in precalculus, extrema are often identified by analyzing the function's formula or graph.