Real-Valued Functions in Mathematics

Questions and Answers

What is a sequence of real numbers mathematically defined as?

A function from the integers to the real numbers.

A function from the set of positive natural numbers to the rationals.

A function from the set of positive natural numbers to the real numbers. (correct)

A collection of real numbers without any specific order.

Which of the following sequences is an example of a converging sequence?

an = (-1)^n

an = 2n

an = 1/n (correct)

an = n^2

What property must a monotonic sequence have to ensure convergence in the context of real numbers?

It must be decreasing.

It must be bounded. (correct)

It must vary between negative and positive values.

It must be strictly increasing.

When visualizing a sequence as a function, what does the graph represent?

The terms of the sequence plotted against their index values. Signup and view all the answers

What is the first term of the Fibonacci sequence as defined in the document?

1 Signup and view all the answers

What distinguishes the functions f(x) = x² + 5 and g(x) = x² + 5 when considering their domains and codomains?

They differ in their codomain, making them distinct functions. Signup and view all the answers

Study Notes

Functions and Their Properties

A function f maps elements from set A (domain) to set B (codomain), represented as f: A → B, with the graph being a subset of A × B.
Real-valued functions have a codomain of real numbers R, for instance, f: A → R.
Examples illustrate that weight of a dog (f(a)) relates to set A (dogs) and IQ of students relates to set B (students in MA 105).

Functions on Intervals

Functions can be defined on specific intervals like f: [0, 1] → R with f(x) = x² + 5, and g: [0, 1] → (3, 10) with g(x) = x² + 5.
Distinction is made between functions f and g due to differing codomains.

Absolute Value Function

The absolute value function f: R → R is defined as f(x) = |x|.
Key properties include:
- |x| ≥ 0, with equality only when x = 0 (range: [0, ∞)).
- |x| = |−x|.
- |xy| = |x||y|.
- −|x| ≤ x ≤ |x|.
- Triangle inequality: |x + y| ≤ |x| + |y|.

Sine and Cosine Functions

The functions f(x) = sin x and f(x) = cos x are periodic with values oscillating between -1 and 1.
The graph of f(x) = sin(1/x) for x > 0 shows rapid oscillation as x approaches zero.

Sequences

A sequence is a function f: N+ → R mapping positive natural numbers to real numbers, denoting terms as {an}.
Examples of sequences include:
- an = 1/n (converges to 0).
- an = n (increasing).
- an = (−1)^n oscillates between -1 and 1.
- an = n² (growing quadratically).
- an = 2 (constant sequence).
- an = 2n (exponential growth).
- The Fibonacci sequence defined recursively.

Visualizing Sequences

Sequences can be visualized on a number line or as graphs, marking individual terms like a1, a2, a3, etc.

Composite Functions

Composite functions involve two functions, where g ◦ f is defined as (g ◦ f)(a) := g(f(a)), creating a new function mapping elements from A to C.

Importance of Functions

Functions are crucial for relating sets, especially in real-valued contexts, encompassing bounded, monotone, and convex functions.

Description

This quiz explores the concept of functions between sets, specifically focusing on real-valued functions. It includes definitions, examples, and illustrations to help understand how functions map elements from one set to another. Test your knowledge on this essential aspect of mathematics!