Functions and Their Properties

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

Which of the following statements accurately describes the concept of a well-defined or single-valued function in the context of the provided text?

A function is well-defined if every element in the codomain has at least one corresponding element in the domain.

A function is well-defined if the mapping between elements in the domain and codomain is consistent across multiple occurrences of the same input value.

A function is well-defined if each element in the domain maps to a single, unique element in the codomain. (correct)

A function is well-defined if the domain and codomain have the same number of elements.

Consider the function f defined as f(x) = x^2. Based on the provided text, what would be the range of this function if its domain is restricted to the set of natural numbers, {1, 2, 3, ...}?

All non-negative real numbers

All positive integers (correct)

All real numbers

All even integers

In the example f = {(x, y): y = x^2}, the text states that 'any real number x is mapped to its square.' Which of the following statements best describes this mapping in terms of function notation?

For every x in the domain, there exists exactly one y in the codomain such that y = x^2. (correct)

For every x in the domain, there exists at least one y in the codomain such that y = x^2.

For every y in the codomain, there exists exactly one x in the domain such that y = x^2.

For every y in the codomain, there exists at least one x in the domain such that y = x^2.

Which of the following accurately describes the relationship between the independent and dependent variables in the context of functions, as presented in the text?

The dependent variable's value is directly determined by the value of the independent variable. (A) Signup and view all the answers

In the context of functions, what is the significance of the statement 'f is a function from ℝ into ℝ'?

It implies f is a function whose domain and codomain are both sets of real numbers. (B) Signup and view all the answers

Given the function f defined as f(1) = 1, f(2) = 6, f(3) = 8, f(4) = 8, which of the following statements is true based on the provided text?

f is a function because each element in the domain has a unique image. (C) Signup and view all the answers

Consider the function f defined as f(x) = 2x + 1. Which of the following statements accurately describes the relationship between the domain and range of this function, based on the provided text?

The domain and range of f are not related in any specific way. (D) Signup and view all the answers

If a function f: A -> B is onto and A is a finite set, which of the following statements is always true?

f is one-to-one. (A) Signup and view all the answers

Consider the function f(n) = 2n for all n ∈ Z. Which of the following statements about f is true?

f is one-to-one, but not onto. (B) Signup and view all the answers

Given a function f(x) = x^2, what is the inverse function f^(-1)(x)?

The function does not have an inverse. (D) Signup and view all the answers

Let A = {1, 2, 3, 4} and B = {5, 6, 7, 8}. Which of the following functions from A to B is a one-to-one correspondence?

f(1) = 5, f(2) = 6, f(3) = 7, f(4) = 8 (A) Signup and view all the answers

Which of the following is not a necessary condition for a function to have an inverse?

The function must be defined on a finite set. (C) Signup and view all the answers

Let g(x) = 2x + 1. What is the inverse function, g^(-1)(x)?

(x + 1) / 2 (A) Signup and view all the answers

Given f(x) = x^3 + 1, what is the range of the inverse function f^(-1)(x)?

All real numbers (C) Signup and view all the answers

Let h(x) = 4x - 2. Which of the following statements about the inverse function h^(-1)(x) is true?

h^(-1)(x) is a linear function with a positive slope. (A) Signup and view all the answers

Consider the function f(x) = 1/x for x ≠ 0. Which of the following statements is true about its inverse f^(-1)(x)?

f^(-1)(x) = 1/x for x ≠ 0 (C) Signup and view all the answers

A function f: A -> B is defined as f(x) = x^2 + 1 for all x ∈ A. If A = {1, 2, 3} and B = {2, 5, 10}, which of the following statements about f is true?

f is one-to-one but not onto. (D) Signup and view all the answers

What is the value of $((-2)^3)^{\frac{1}{6}}$ simplified to its lowest form?

$-2$ (B) Signup and view all the answers

Simplify the expression: $16^{\frac{3}{4}}$

$8$ (A) Signup and view all the answers

Express $\sqrt[3]{x^5}$ in terms of a rational exponent.

$x^{\frac{5}{3}}$ (D) Signup and view all the answers

Which of the following is equivalent to $a^{-\frac{m}{n}}$ where 'a' is a non-zero constant?

$\frac{1}{a^{\frac{m}{n}}}$ (D) Signup and view all the answers

Consider the expression $(x^2y^3)^{\frac{1}{2}}$. What is the simplified equivalent in terms of x and y?

$xy^{\frac{3}{2}}$ (D) Signup and view all the answers

Given that $f(x) = x^2 + x$ and $g(x) = \frac{1}{x+3}$ , determine the value of $(g \circ f)(1)$ .

$\frac{1}{4}$ (D) Signup and view all the answers

For the functions $f(x) = x^3 + 2$ and $g(x) = \frac{2}{x-1}$, find the domain of $(f \circ g)(x)$ .

x ≠ 1, x ≠ 3 (A) Signup and view all the answers

Let $f(x) = x^2$ and $g(x) = x$. Find $(f \circ g)(x)$ and its domain.

$(f \circ g)(x)=x^2$, domain: all real numbers (B) Signup and view all the answers

For the functions $f(x) = 5x - 3$ and $(f \circ g)(x) = 2x + 7$, find $g(x)$.

$g(x) = \frac{2}{5}x - 8$ (D) Signup and view all the answers

If $f(x) = 2x + 1$ and $(f \circ g)(x) = 3x - 1$, find $g(x)$.

$g(x) = \frac{3}{2}x - 2$ (E) Signup and view all the answers

For the function $f(x) = \frac{x-1}{x+1}$, find the value of $f(2x)$.

$\frac{2x-1}{2x+1}$ (B) Signup and view all the answers

Find two functions $f$ and $g$ such that the given function $h(x) = (f \circ g)(x)$ is equivalent to $h(x) = (x+3)^3$.

$f(x) = x^3$, $g(x) = x+3$ (A) Signup and view all the answers

Let $f(x) = 4x - 3$, $g(x) = \frac{1}{x}$, and $h(x) = x^2 - x$. Find $f(g(h(3)))$.

$\frac{1}{3}$ (D) Signup and view all the answers

Given that $f(x) = 4x - 3$, $g(x) = \frac{1}{x}$, and $h(x) = x^2 - x$, find $f(1) \cdot g(2) \cdot h(3)$.

-2 (F) Signup and view all the answers

Signup and view all the answers

Study Notes

Chapter 3: Functions

Everyday life involves relationships between sets. Examples include license plates to automobiles and circumference to circles.
Relationships between sets need mathematical precision.
Key skills students will gain are understanding relations and functions, determining domain and range of relations and functions, finding inverses of relations, defining and performing operations on polynomial and rational functions, applying theorems to find polynomial function zeros, sketching graphs of various functions, and understanding and applying properties of various types of real-valued functions.

Section 3.1: Review of Relations and Functions

Cartesian product of two sets A and B (A x B) is the set of all ordered pairs (a,b) where a ∈ A and b ∈ B.
Ordered pair (a,b) is equal to ordered pair (c,d) if and only if a=c and b=d.
A relation from set A to set B is any subset of A x B.
a is R-related to b (written aRb) if (a,b) ∈ R.
The domain of R is the set of all first coordinates of the ordered pairs in R, and the range of R is the set of all second coordinates.
If R is a relation from A to itself, then R is a relation on A.
A function is a relation in which each element of the domain corresponds to exactly one element of the range.
For f : A → B to be a function, Dom(f) = A and each a ∈ A must map to at most one element in B.
A function is a correspondence that assigns to each element of a set A exactly one element of a set B. For example, if f is a function from a set A to a set B, and (a,b) ∈ f, then each value of a must have only one corresponding value of b.
A relation f from A into B is called a function from A into B iff Dom(f) = A, and no element of A is mapped by f to more than one element of B.
If (x, y) ∈ f, then y is called the image of x under f and x is called the pre-image of y under f.
The input values form the domain, and output values form the range of a function.

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Description

This quiz explores the fundamental concepts of functions, including single-valued functions, ranges, and the relationships between independent and dependent variables. You will analyze examples and definitions to deepen your understanding of mathematical functions.