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
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter9: Quadratic Functions And Equations
Section9.9: Combining Functions
Problem 51HP
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
Glencoe Algebra 1, Student Edition, 9780079039897…
Algebra
ISBN:
9780079039897
Author:
Carter
Publisher:
McGraw Hill
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
Glencoe Algebra 1, Student Edition, 9780079039897…
Algebra
ISBN:
9780079039897
Author:
Carter
Publisher:
McGraw Hill
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
Holt Mcdougal Larson Pre-algebra: Student Edition…
Algebra
ISBN:
9780547587776
Author:
HOLT MCDOUGAL
Publisher:
HOLT MCDOUGAL
Algebra: Structure And Method, Book 1
Algebra
ISBN:
9780395977224
Author:
Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:
McDougal Littell
Big Ideas Math A Bridge To Success Algebra 1: Stu…
Algebra
ISBN:
9781680331141
Author:
HOUGHTON MIFFLIN HARCOURT
Publisher:
Houghton Mifflin Harcourt