Chapter 1: Functions and Representations

Flashcards covering key concepts from the lecture notes on functions, representations, domain/range, graphs, and example problems.

What is a transcendental function?

A function that cannot be expressed using a finite combination of algebraic operations (addition, subtraction, multiplication, division, exponentiation, and root extraction).

Give examples of transcendental functions.

Trigonometric functions (sin x, cos x, tan x), the exponential function e^x, and the natural logarithm ln x.

What is the fundamental object studied in calculus?

Functions—their graphs and how they transform or combine.

List the four representations of a function.

Equation, table, graph, and words.

Define the domain of a function.

The set of input values for which the function is defined.

Define the range of a function.

The set of all possible values f(x) as x varies over the domain.

What is an independent variable?

A value in the domain of the function, typically denoted x, that serves as input.

What is a dependent variable?

The output value, usually denoted f(x) or y, that depends on the input.

What does increasing on an interval mean?

If x1 < x2 in the interval, then f(x1) < f(x2).

What does decreasing on an interval mean?

If x1 < x2 in the interval, then f(x1) > f(x2).

What is the Vertical Line Test?

A graph represents a function if every vertical line intersects the graph at most once.

What is an even function?

A function f with f(-x) = f(x) for all x in its domain; graph is symmetric about the y-axis.

What is an odd function?

A function f with f(-x) = -f(x) for all x in its domain; graph is symmetric about the origin.

What is a piecewise defined function?

A function defined by different formulas on different parts of its domain.

Domain and range for A = π r^2 as a function of r?

Domain (0, ∞); Range (0, ∞).

What does f(x) represent on a graph?

The height (y-coordinate) of the graph at the point x, i.e., the value of the function at x.

How is a function represented by a graph?

As the set of points (x, f(x)) with x in the domain.

What is an arrow diagram in the context of functions?

A diagram showing how each x in the domain maps to f(x) in the codomain.

In a graph, what do the x-axis and y-axis represent for a function?

The x-axis represents the domain; the y-axis represents the range (values of f(x)).

What is the domain and range for Example 1's graph?

Domain [0, 7]; Range [-2, 4].

What is f(1) in Example 1?

f(1) = 3.

What is f(5) in Example 1?

f(5) ≈ -0.7.

What is the domain of the function in Example 1's graph?

0 ≤ x ≤ 7 (domain [0, 7]).

In the open-top box problem, express the base and side material costs per square meter.

Base costs $10 per m^2; Sides cost $6 per m^2.

What is the base area for the container problem with width w (length = 2w)?

Base area = 2 w^2.

If the volume is 10 m^3, how is height h expressed in terms of w?

h = 10 / (2 w^2) = 5 / w^2.

What is the total cost function C(w) for the container problem after substitution?

C(w) = 20 w^2 + 180 / w.

What is the standard algebraic expression for the area of a circle as a function of radius?

A = π r^2.

What is the difference quotient and what does it represent?

The difference quotient is [f(a+h) - f(a)]/h (with h ≠ 0) and it represents the average rate of change of f between a and a+h.

What is a piecewise function example rule near x = -1 in Example 7?

If x ≤ -1, f(x) = 1 − x; if x > -1, f(x) = x^2.

How is the graph of a piecewise function typically drawn around the breakpoint?

Plot each piece on its respective interval and use open/closed dots to indicate which values are included at the breakpoint.

What is the rule for an even function's symmetry?

The graph is symmetric about the y-axis because f(-x) = f(x).

What is the rule for an odd function's symmetry?

The graph is symmetric about the origin because f(-x) = -f(x).

What is the defining property of an increasing function on an interval I?

If x1 < x2 in I, then f(x1) < f(x2).

What is the defining property of a decreasing function on an interval I?

If x1 < x2 in I, then f(x1) > f(x2).