Let f(x) be a function with domain R. (a) Show that E(x) = f(x) + f(-x) is an even function. (Simplify your answers completely.) E(x) = f(x) + f(-x) = Since E(-x) = E O(x) = f(x) = f(-x) = Since O(-x) = -0 E(-x) = f(-x)+ +(-([ = f(-x) + f = E(x) (b) Show that O(x) = f(x) = f(-x) is an odd function. (Simplify your answers completely.) 0(-x) = f( )-f(-(-x)) E is an even function. = f(-x) = f(x) - [F(X) - F = -0(x) 21 E(X) + 12/10 (x) = 21 [ R(x) + f( [ = 1/2 [1
Let f(x) be a function with domain R. (a) Show that E(x) = f(x) + f(-x) is an even function. (Simplify your answers completely.) E(x) = f(x) + f(-x) = Since E(-x) = E O(x) = f(x) = f(-x) = Since O(-x) = -0 E(-x) = f(-x)+ +(-([ = f(-x) + f = E(x) (b) Show that O(x) = f(x) = f(-x) is an odd function. (Simplify your answers completely.) 0(-x) = f( )-f(-(-x)) E is an even function. = f(-x) = f(x) - [F(X) - F = -0(x) 21 E(X) + 12/10 (x) = 21 [ R(x) + f( [ = 1/2 [1
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Question
I need help with this problem and explanation to this problem.
![Let f(x) be a function with domain R.
(a) Show that E(x) = f(x) + f(-x) is an even function. (Simplify your answers completely.)
E(x) = f(x) + f(-x) => E(-x) = f(-x) + f
= f(-x) + f
= E(x)
Since E(-x) = E
(b) Show that O(x) = f(x) − f(-x) is an odd function. (Simplify your answers completely.)
= f(C
O(x) = f(x) = f(-x) =
Since O(-x) = -ol
0(-x) =
12 E(X) + 12/10 (x) = ²/1 [F(x) + f ( [
1/2/ [F(x) + f ( [
=
== /[2(x)]
x) = -1/-(2 ²
E is an even function.
O f(x)=
= f(-x) = f(x)
= - [F(x) - F( [
= -0(x)
(c) Prove that every function f(x) can be written as a sum of an even function and an odd function.
For any function f with domain R, define functions E(x) = f(x) + f(-x) and O(x) = f(x) - f(-x), as in parts (a) and (b). Then Eis ---Select---
Of(x) = 1/(2*
-f(-(-x))
O is an odd function.
(d) Using parts (a), (b), and (c), express the function f(x) = 2* + (x − 3)² as a sum of an even function and an odd function.
´ + (x − 3)² + 2¯× + (x + 3)²) + ½¹⁄(2* + (x − 3)² − 2¯× − (x+3)²)
+ (x
]] + [-(-x]
] ) + F(x) = f(-x)]
+ (x − 3)² + 2¯* + (x + 3)²) − ½ (2* + (x − 3)² − 2¯× − (x + 3)²)
+
(x
○ F(x) = ¹/(2* + (x − 3)² − 2¯× − (x + 3)²) + ¹⁄(2× + (x − 3)² − 2¯× − (x + 3)²)
O f(x) = (2x + (x − 3)² + 2x + (x + 3)²) − (2* + (x − 3)² − 2¯* — (x + 3)²)
○ f(x) = (2x + (x − 3)² + 2¯* + (x + 3)²) + (2* + (x − 3)² − 2¯× − (x + 3)²)
○ f(x) = (2x + (x − 3)² - 2¯* − (x + 3)²) + (2* + (x − 3)² − 2¯* − (x + 3)²)
20 is
O is --Select---✓, and we show that f(x) = E(x) + 10(x). (Simplify your answers completely.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4e135d5c-f867-4afd-9f0d-b505f4f19664%2Fff775664-74e6-4db6-ab2a-140cfb29eab2%2Fxmp3wv_processed.png&w=3840&q=75)
Transcribed Image Text:Let f(x) be a function with domain R.
(a) Show that E(x) = f(x) + f(-x) is an even function. (Simplify your answers completely.)
E(x) = f(x) + f(-x) => E(-x) = f(-x) + f
= f(-x) + f
= E(x)
Since E(-x) = E
(b) Show that O(x) = f(x) − f(-x) is an odd function. (Simplify your answers completely.)
= f(C
O(x) = f(x) = f(-x) =
Since O(-x) = -ol
0(-x) =
12 E(X) + 12/10 (x) = ²/1 [F(x) + f ( [
1/2/ [F(x) + f ( [
=
== /[2(x)]
x) = -1/-(2 ²
E is an even function.
O f(x)=
= f(-x) = f(x)
= - [F(x) - F( [
= -0(x)
(c) Prove that every function f(x) can be written as a sum of an even function and an odd function.
For any function f with domain R, define functions E(x) = f(x) + f(-x) and O(x) = f(x) - f(-x), as in parts (a) and (b). Then Eis ---Select---
Of(x) = 1/(2*
-f(-(-x))
O is an odd function.
(d) Using parts (a), (b), and (c), express the function f(x) = 2* + (x − 3)² as a sum of an even function and an odd function.
´ + (x − 3)² + 2¯× + (x + 3)²) + ½¹⁄(2* + (x − 3)² − 2¯× − (x+3)²)
+ (x
]] + [-(-x]
] ) + F(x) = f(-x)]
+ (x − 3)² + 2¯* + (x + 3)²) − ½ (2* + (x − 3)² − 2¯× − (x + 3)²)
+
(x
○ F(x) = ¹/(2* + (x − 3)² − 2¯× − (x + 3)²) + ¹⁄(2× + (x − 3)² − 2¯× − (x + 3)²)
O f(x) = (2x + (x − 3)² + 2x + (x + 3)²) − (2* + (x − 3)² − 2¯* — (x + 3)²)
○ f(x) = (2x + (x − 3)² + 2¯* + (x + 3)²) + (2* + (x − 3)² − 2¯× − (x + 3)²)
○ f(x) = (2x + (x − 3)² - 2¯* − (x + 3)²) + (2* + (x − 3)² − 2¯* − (x + 3)²)
20 is
O is --Select---✓, and we show that f(x) = E(x) + 10(x). (Simplify your answers completely.)
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 1 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning