MATH 1111 – College Algebra Chapter 1/Module 1 Supplemental Material: Graphing and Evaluating Piecewise Functions

Flashcards covering domain and range of common functions, interval notation, piecewise functions, graphing guidelines, and endpoint conventions.

What is the domain of the constant function f(x) = c?

All real numbers.

What is the range of the constant function f(x) = c?

The singleton set {c}.

In interval notation, how is the range of a constant function written?

[c, c] (the interval that begins and ends at c).

For the identity function f(x) = x, what is the domain?

All real numbers.

For the identity function f(x) = x, what is the range?

All real numbers.

For the absolute value function f(x) = |x|, what is the domain?

All real numbers.

For the absolute value function f(x) = |x|, what is the range?

y ≥ 0 (nonnegative real numbers).

For the quadratic function f(x) = x^2, what is the domain?

All real numbers.

For the quadratic function f(x) = x^2, what is the range?

y ≥ 0 (nonnegative real numbers).

For the cubic function f(x) = x^3, what is the domain?

All real numbers.

For the cubic function f(x) = x^3, what is the range?

All real numbers.

For the reciprocal function f(x) = 1/x, what is the domain?

All real numbers except x = 0.

For the reciprocal function f(x) = 1/x, what is the range?

All real numbers except y = 0.

For the reciprocal squared function f(x) = 1/x^2, what is the domain?

All real numbers except x = 0.

For the reciprocal squared function f(x) = 1/x^2, what is the range?

y > 0 (positive only).

For the square root function f(x) = √x, what is the domain?

x ≥ 0.

For the square root function f(x) = √x, what is the range?

y ≥ 0.

For the cube root function f(x) = x^(1/3), what is the domain?

All real numbers.

For the cube root function f(x) = x^(1/3), what is the range?

All real numbers.

What are the general steps to determine the domain and range from a formula?

1) Exclude inputs that cause division by zero; 2) Exclude inputs with nonreal outputs; 3) Use valid inputs to determine the range; 4) Confirm with graph or table.

What is a piecewise function?

A function defined by more than one formula over different parts of the domain.

How can the absolute value function be expressed as a piecewise function?

f(x) = x if x ≥ 0; f(x) = -x if x < 0.

Give a real-world example of a piecewise function.

Tax brackets or volume discounts (e.g., different rules apply beyond thresholds).

What are the main steps when graphing a piecewise function?

1) Indicate the interval boundaries on the x-axis; 2) Graph each piece on its interval using the corresponding equation; 3) Do not graph two pieces on a single interval.

How are endpoints indicated when graphing a piecewise function?

Open circles for endpoints not included; closed circles for endpoints included.

What is the vertical line test?

A graph passes the vertical line test if every vertical line intersects the graph at most once, ensuring a function.

What is a typical procedure for evaluating a piecewise function at specific x-values?

Determine which piece applies for each x, compute the corresponding output, then optionally sketch the graph.

What are the interval notations for x > a and x ≥ a?

(a, ∞) and [a, ∞) respectively.

What are the interval notations for x < a and x ≤ a?

(-∞, a) and (-∞, a] respectively.

What are the interval notations for a < x < b and a ≤ x < b?

(a, b) and [a, b) respectively.

What are the interval notations for a < x ≤ b and a ≤ x ≤ b?

(a, b] and [a, b] respectively.