Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 1: Problem 39
\(39-44=\) Find the functions (a) \(f \circ g,(\mathrm{b}) g \circ f,(\mathrm{c}) f \circ f,\) and (d) \(g \circ g\) and their domains. $$f(x)=x^{2}-1, \quad g(x)=2 x+1$$
Short Answer
Expert verified
All compositions have domain \( \mathbb{R} \).
Step by step solution
01
Understanding Composition of Functions
To find compositions like \( f \circ g \), \( g \circ f \), \( f \circ f \), and \( g \circ g \), you need to replace the input variable of the first function with the second function. For example, for \( f \circ g(x) \), substitute \( g(x) \) into \( f(x) \).
02
Find \( f \circ g(x) \)
To find \( f \circ g \), substitute \( g(x) = 2x + 1 \) into \( f(x) = x^2 - 1 \). This gives \( f(g(x)) = (2x + 1)^2 - 1 = 4x^2 + 4x + 1 - 1 = 4x^2 + 4x \). The domain of \( f \circ g \) is all real numbers, \( \mathbb{R} \).
03
Find \( g \circ f(x) \)
For \( g \circ f \), substitute \( f(x) = x^2 - 1 \) into \( g(x) = 2x + 1 \). This gives \( g(f(x)) = 2(x^2 - 1) + 1 = 2x^2 - 2 + 1 = 2x^2 - 1 \). The domain of \( g \circ f \) is also all real numbers, \( \mathbb{R} \).
04
Find \( f \circ f(x) \)
For \( f \circ f \), substitute \( f(x) = x^2 - 1 \) into the same function \( f(x) = x^2 - 1 \). This gives \( f(f(x)) = (x^2 - 1)^2 - 1 = x^4 - 2x^2 + 1 - 1 = x^4 - 2x^2 \). The domain of \( f \circ f \) is all real numbers, \( \mathbb{R} \).
05
Find \( g \circ g(x) \)
For \( g \circ g \), substitute \( g(x) = 2x + 1 \) into itself \( g(x) = 2x + 1 \). This gives \( g(g(x)) = 2(2x + 1) + 1 = 4x + 2 + 1 = 4x + 3 \). The domain of \( g \circ g \) is all real numbers, \( \mathbb{R} \).
06
Conclusion
All compositions \( f \circ g(x) = 4x^2 + 4x \), \( g \circ f(x) = 2x^2 - 1 \), \( f \circ f(x) = x^4 - 2x^2 \), and \( g \circ g(x) = 4x + 3 \) have real numbers, \( \mathbb{R} \), as their domains.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Function Notation
Function notation is a simple yet powerful way to write and understand mathematical expressions that define relationships between variables. In function notation, a function named "f" with input variable "x" is written as \( f(x) \). This notation helps us indicate what number we are substituting into a function and makes it easier to communicate complex calculations or compositions.
- The expression \( f(x) \) represents the value of the function when the input is \( x \).
- It allows mathematicians to provide clear instructions for what calculations should be performed on a given input.
- Function notation can handle a variety of function types, including linear, quadratic, or even more complex polynomial functions.
Domain of a Function
The domain of a function is the complete set of possible input values (usually \( x \)) that the function can accept without causing mathematical errors. Understanding the domain is crucial because it defines the boundaries for function use.
- For a basic linear function, like \( g(x) = 2x + 1 \), the domain is all real numbers \( \mathbb{R} \) since you can plug any real number into the function without encountering issues.
- With quadratic functions like \( f(x) = x^2 - 1 \), the domain remains all real numbers \( \mathbb{R} \) because squaring any real number results in a real output.
- Domains become more complicated with functions involving division by zero, square roots of negative numbers, or logarithms of non-positive numbers, which are undefined in real numbers.
Quadratic Function
A quadratic function is a type of polynomial function characterized by having the highest power of the variable \( x \) as two (squared). Its general form is \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \).
- Quadratic functions typically create a parabolic shape when graphed. The parabola can open upwards or downwards depending on whether \( a \) is positive or negative, respectively.
- These functions may have roots, or x-intercepts, where the function value becomes zero.
- They have a vertex, either a maximum or minimum point, depending on the parabola's direction.
Polynomial Function
Polynomial functions are expressions that involve sums of powers of \( x \) with constant coefficients. The general form of a polynomial function is \( f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0 \), where \( n \) is a non-negative integer and \( a_n, \ldots, a_0 \) are constants.
- The degree of the polynomial function is the highest power of \( x \) with a non-zero coefficient.
- Polynomial functions are continuous and smooth, meaning they do not have breaks or sharp corners.
- They include a range of functions from simple linear (degree 1) functions to higher-order functions like quadratic (degree 2), cubic (degree 3), and so on.
