Single Variable Calculus: Concepts and Contexts, Enhanced Edition

4th Edition

ISBN: 9781337687805

Author: James Stewart

Publisher: Cengage Learning

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Question

Chapter 1.1, Problem 73E

To determine

Whether the function $f+g$ is even if f and g are even; whether the function $f+g$ is odd if f and g are odd; whether the function $f+g$ is even or odd if f is even and g is odd.

Expert Solution & Answer

Answer to Problem 73E

The function $f+g$ is an even function if f and g are even.

The function $f+g$ is an odd function if f and g are odd.

The function $f+g$ neither an even function nor an odd function if f is even and g is odd.

Explanation of Solution

Definition used:

If $f(x)=f(−x)$ , then the function $f(x)$ is said to be an even function.

If $f(x)≠f(−x)$ , then the function $f(x)$ is not an even function.

If $f(−x)=−f(x)$ , then the function $f(x)$ is said to be an odd function.

If $f(−x)≠−f(x)$ , then the function $f(x)$ is not an odd function.

Calculation:

Section (i)

If f and g are even functions, then by the definition, $f(x)=f(−x)$ and $g(x)=g(−x)$ .

Recall the fact that, $(f+g)(−x)=f(−x)+g(−x)$ (1)

As f and g are even functions, substitute $f(−x)=f(x)$ and $g(−x)=g(x)$ in equation (1) as follows.

$(f+g)(−x)=f(−x)+g(−x)=f(x)+g(x)=(f+g)(x)$

Therefore, $f+g$ is an even function as it satisfies the definition of even function, $(f+g)(−x)=(f+g)(x)$ .

Section (ii)

If f and g are odd functions, then by definition, $f(−x)=−f(x)$ and $g(−x)=−g(x)$ .

As f and g are odd functions, substitute $f(−x)=−f(x)$ and $g(−x)=−g(x)$ in equation (1) as follows.

$(f+g)(−x)=−f(x)−g(x)=−(f(x)+g(x))=−((f+g)(x))$

Therefore, $f+g$ is an odd function as it satisfies the definition of odd function, $(f+g)(−x)=−((f+g)(x))$ .

Section (iii)

Given that, f is an even function and g is an odd function.

Then by the definition, $f(x)=f(−x)$ and $g(−x)=−g(x)$ .

Therefore, substitute $f(x)=f(−x)$ and $g(−x)=−g(x)$ in equation (1),

$(f+g)(−x)=f(−x)+g(−x)=f(x)−g(x)$

Observe that the function $f(x)−g(x)$ neither satisfies the condition of even function nor the condition of odd function.

Therefore, the function $f+g$ is neither an even function nor an odd function if f is even and g is odd.

Chapter 1.1, Problem 72E

Chapter 1.1, Problem 74E

Chapter 1 Solutions

Single Variable Calculus: Concepts and Contexts, Enhanced Edition

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Ch. 1.1 - Prob. 12E Ch. 1.1 - The graph shows the power consumption for a day in...Ch. 1.1 - Prob. 14E Ch. 1.1 - Prob. 15E Ch. 1.1 - Prob. 16E Ch. 1.1 - Sketch the graph of the amount of a particular...Ch. 1.1 - You place a frozen pie in an oven and bake it for...Ch. 1.1 - Prob. 19E Ch. 1.1 - Prob. 20E Ch. 1.1 - Prob. 21E Ch. 1.1 - Prob. 22E Ch. 1.1 - Prob. 23E Ch. 1.1 - Prob. 24E Ch. 1.1 - Prob. 25E Ch. 1.1 - Prob. 26E Ch. 1.1 - Prob. 27E Ch. 1.1 - Prob. 28E Ch. 1.1 - Prob. 29E Ch. 1.1 - Prob. 30E Ch. 1.1 - Prob. 31E Ch. 1.1 - Prob. 32E Ch. 1.1 - Prob. 33E Ch. 1.1 - Prob. 34E Ch. 1.1 - Prob. 35E Ch. 1.1 - Prob. 36E Ch. 1.1 - Prob. 37E Ch. 1.1 - Prob. 38E Ch. 1.1 - Prob. 39E Ch. 1.1 - Prob. 40E Ch. 1.1 - Prob. 41E Ch. 1.1 - Prob. 42E Ch. 1.1 - Prob. 43E Ch. 1.1 - Prob. 44E Ch. 1.1 - Prob. 45E Ch. 1.1 - Prob. 46E Ch. 1.1 - Prob. 47E Ch. 1.1 - Prob. 48E Ch. 1.1 - Find an expression for the function whose graph is...Ch. 1.1 - Prob. 50E Ch. 1.1 - Prob. 51E Ch. 1.1 - Prob. 52E Ch. 1.1 - Prob. 53E Ch. 1.1 - Prob. 54E Ch. 1.1 - Prob. 55E Ch. 1.1 - Prob. 56E Ch. 1.1 - Prob. 57E Ch. 1.1 - A Norman window has the shape of a rectangle...Ch. 1.1 - Prob. 59E Ch. 1.1 - Prob. 60E Ch. 1.1 - Prob. 61E Ch. 1.1 - Prob. 62E Ch. 1.1 - Prob. 63E Ch. 1.1 - Prob. 64E Ch. 1.1 - Prob. 65E Ch. 1.1 - Prob. 66E Ch. 1.1 - Prob. 67E Ch. 1.1 - Prob. 68E Ch. 1.1 - Prob. 69E Ch. 1.1 - Prob. 70E Ch. 1.1 - Prob. 71E Ch. 1.1 - Prob. 72E Ch. 1.1 - Prob. 73E Ch. 1.1 - Prob. 74E Ch. 1.2 - Classify each function as a power function, root...Ch. 1.2 - Prob. 2E Ch. 1.2 - Prob. 3E Ch. 1.2 - Prob. 4E Ch. 1.2 - Prob. 5E Ch. 1.2 - Prob. 6E Ch. 1.2 - Prob. 7E Ch. 1.2 - Prob. 8E Ch. 1.2 - Prob. 9E Ch. 1.2 - Prob. 10E Ch. 1.2 - Prob. 11E Ch. 1.2 - The manager of a weekend flea market knows from...Ch. 1.2 - Prob. 13E Ch. 1.2 - Prob. 14E Ch. 1.2 - Biologists have noticed that the chirping rate of...Ch. 1.2 - Prob. 16E Ch. 1.2 - Prob. 17E Ch. 1.2 - Prob. 18E Ch. 1.2 - Prob. 19E Ch. 1.2 - Prob. 20E Ch. 1.2 - Prob. 21E Ch. 1.2 - Prob. 22E Ch. 1.2 - Prob. 23E Ch. 1.2 - Prob. 24E Ch. 1.2 - Prob. 25E Ch. 1.2 - Prob. 26E Ch. 1.3 - Suppose the graph of f is given. Write equations...Ch. 1.3 - Prob. 2E Ch. 1.3 - The graph of y=f(x) is given. Match each equation...Ch. 1.3 - Prob. 4E Ch. 1.3 - Prob. 5E Ch. 1.3 - Prob. 6E Ch. 1.3 - Prob. 7E Ch. 1.3 - Prob. 8E Ch. 1.3 - Prob. 9E Ch. 1.3 - Prob. 10E Ch. 1.3 - Prob. 11E Ch. 1.3 - Prob. 12E Ch. 1.3 - Prob. 13E Ch. 1.3 - Prob. 14E Ch. 1.3 - Prob. 15E Ch. 1.3 - Prob. 16E Ch. 1.3 - Prob. 17E Ch. 1.3 - Prob. 18E Ch. 1.3 - Prob. 19E Ch. 1.3 - Prob. 20E Ch. 1.3 - Prob. 21E Ch. 1.3 - Prob. 22E Ch. 1.3 - Prob. 23E Ch. 1.3 - Prob. 24E Ch. 1.3 - The city of New Orleans is located at latitude...Ch. 1.3 - Prob. 26E Ch. 1.3 - Prob. 27E Ch. 1.3 - Prob. 28E Ch. 1.3 - Prob. 29E Ch. 1.3 - Prob. 30E Ch. 1.3 - Prob. 31E Ch. 1.3 - Prob. 32E Ch. 1.3 - Prob. 33E Ch. 1.3 - Prob. 34E Ch. 1.3 - Prob. 35E Ch. 1.3 - Prob. 36E Ch. 1.3 - Prob. 37E Ch. 1.3 - Prob. 38E Ch. 1.3 - Prob. 39E Ch. 1.3 - Prob. 40E Ch. 1.3 - Prob. 41E Ch. 1.3 - Prob. 42E Ch. 1.3 - Prob. 43E Ch. 1.3 - Prob. 44E Ch. 1.3 - Express the function in the form f g. 47. v(t) =...Ch. 1.3 - Prob. 46E Ch. 1.3 - Prob. 47E Ch. 1.3 - Prob. 48E Ch. 1.3 - Prob. 49E Ch. 1.3 - Prob. 50E Ch. 1.3 - Prob. 51E Ch. 1.3 - Use the given graphs of f and g to estimate the...Ch. 1.3 - Prob. 53E Ch. 1.3 - Prob. 54E Ch. 1.3 - A ship is at a speed of 30km/h parallel to a...Ch. 1.3 - Prob. 56E Ch. 1.3 - Prob. 57E Ch. 1.3 - The Heaviside function defined in Exercise 59 can...Ch. 1.3 - Prob. 59E Ch. 1.3 - Prob. 60E Ch. 1.3 - Prob. 61E Ch. 1.3 - Prob. 62E Ch. 1.3 - Prob. 63E Ch. 1.3 - Prob. 64E Ch. 1.4 - Prob. 1E Ch. 1.4 - Prob. 2E Ch. 1.4 - Prob. 3E Ch. 1.4 - Prob. 4E Ch. 1.4 - Prob. 5E Ch. 1.4 - Prob. 6E Ch. 1.4 - Prob. 7E Ch. 1.4 - Prob. 8E Ch. 1.4 - Prob. 9E Ch. 1.4 - Prob. 10E Ch. 1.4 - Prob. 11E Ch. 1.4 - Prob. 12E Ch. 1.4 - Prob. 13E Ch. 1.4 - Prob. 14E Ch. 1.4 - Prob. 15E Ch. 1.4 - Prob. 16E Ch. 1.4 - Prob. 17E Ch. 1.4 - Prob. 18E Ch. 1.4 - Prob. 19E Ch. 1.4 - Prob. 20E Ch. 1.4 - Prob. 21E Ch. 1.4 - Prob. 22E Ch. 1.4 - Prob. 23E Ch. 1.4 - Prob. 24E Ch. 1.4 - Prob. 25E Ch. 1.4 - Prob. 26E Ch. 1.4 - Prob. 27E Ch. 1.4 - Prob. 28E Ch. 1.4 - Prob. 29E Ch. 1.4 - Prob. 30E Ch. 1.4 - Prob. 31E Ch. 1.4 - Prob. 32E Ch. 1.4 - Prob. 33E Ch. 1.4 - Prob. 34E Ch. 1.4 - Prob. 35E Ch. 1.4 - Prob. 36E Ch. 1.4 - Prob. 37E Ch. 1.4 - Prob. 38E Ch. 1.5 - Use the Law of Exponents to rewrite and simplify...Ch. 1.5 - Use the Law of Exponents to rewrite and simplify...Ch. 1.5 - Prob. 3E Ch. 1.5 - Use the Law of Exponents to rewrite and simplify...Ch. 1.5 - Prob. 5E Ch. 1.5 - Prob. 6E Ch. 1.5 - Prob. 7E Ch. 1.5 - Prob. 8E Ch. 1.5 - Prob. 9E Ch. 1.5 - Prob. 10E Ch. 1.5 - Prob. 11E Ch. 1.5 - Prob. 12E Ch. 1.5 - Prob. 13E Ch. 1.5 - Prob. 14E Ch. 1.5 - Make a rough sketch of the graph of the function....Ch. 1.5 - Prob. 16E Ch. 1.5 - Prob. 17E Ch. 1.5 - Prob. 18E Ch. 1.5 - Prob. 19E Ch. 1.5 - Prob. 20E Ch. 1.5 - Prob. 21E Ch. 1.5 - Find the exponential function f(x) = Cbx whose...Ch. 1.5 - Prob. 23E Ch. 1.5 - Suppose you are offered a job that lasts one...Ch. 1.5 - Prob. 25E Ch. 1.5 - Prob. 26E Ch. 1.5 - Prob. 27E Ch. 1.5 - Prob. 28E Ch. 1.5 - Prob. 29E Ch. 1.5 - A bacteria culture starts with 500 bacteria and...Ch. 1.5 - The half-life of bismuth-210, 210Bi, is 5 days....Ch. 1.5 - Prob. 32E Ch. 1.5 - Prob. 33E Ch. 1.5 - Prob. 34E Ch. 1.5 - Prob. 35E Ch. 1.5 - Prob. 36E Ch. 1.6 - (a) What is a one-to-one function? 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(a) If...Ch. 1.6 - Prob. 16E Ch. 1.6 - Prob. 17E Ch. 1.6 - Prob. 18E Ch. 1.6 - Prob. 19E Ch. 1.6 - Prob. 20E Ch. 1.6 - Prob. 21E Ch. 1.6 - Prob. 22E Ch. 1.6 - Prob. 23E Ch. 1.6 - Prob. 24E Ch. 1.6 - Find a formula for the inverse of the function....Ch. 1.6 - Prob. 26E Ch. 1.6 - Prob. 27E Ch. 1.6 - Prob. 28E Ch. 1.6 - Prob. 29E Ch. 1.6 - Prob. 30E Ch. 1.6 - Prob. 31E Ch. 1.6 - Prob. 32E Ch. 1.6 - Prob. 33E Ch. 1.6 - Prob. 34E Ch. 1.6 - Prob. 35E Ch. 1.6 - Prob. 36E Ch. 1.6 - Prob. 37E Ch. 1.6 - Prob. 38E Ch. 1.6 - Prob. 39E Ch. 1.6 - Prob. 40E Ch. 1.6 - Prob. 41E Ch. 1.6 - Use Formula 10 to evaluate each logarithm correct...Ch. 1.6 - Prob. 43E Ch. 1.6 - Prob. 44E Ch. 1.6 - Prob. 45E Ch. 1.6 - Prob. 46E Ch. 1.6 - Prob. 47E Ch. 1.6 - Make a rough sketch of the graph of each function....Ch. 1.6 - Solve each equation for x. 51. (a) e74x=6 (b)...Ch. 1.6 - Prob. 50E Ch. 1.6 - Prob. 51E Ch. 1.6 - Prob. 52E Ch. 1.6 - Prob. 53E Ch. 1.6 - Prob. 54E Ch. 1.6 - Prob. 55E Ch. 1.6 - Prob. 56E Ch. 1.6 - Prob. 57E Ch. 1.6 - Prob. 58E Ch. 1.6 - Prob. 59E Ch. 1.6 - Prob. 60E Ch. 1.6 - Prob. 61E Ch. 1.6 - Prob. 62E Ch. 1.7 - Prob. 1E Ch. 1.7 - Prob. 2E Ch. 1.7 - Prob. 3E Ch. 1.7 - Prob. 4E Ch. 1.7 - Prob. 5E Ch. 1.7 - Prob. 6E Ch. 1.7 - Prob. 7E Ch. 1.7 - Prob. 8E Ch. 1.7 - Prob. 9E Ch. 1.7 - Prob. 10E Ch. 1.7 - Prob. 11E Ch. 1.7 - Prob. 12E Ch. 1.7 - Prob. 13E Ch. 1.7 - Prob. 14E Ch. 1.7 - Prob. 15E Ch. 1.7 - Prob. 16E Ch. 1.7 - Prob. 17E Ch. 1.7 - Prob. 18E Ch. 1.7 - Prob. 19E Ch. 1.7 - Prob. 20E Ch. 1.7 - Prob. 21E Ch. 1.7 - Prob. 22E Ch. 1.7 - Prob. 23E Ch. 1.7 - Prob. 24E Ch. 1.7 - Prob. 25E Ch. 1.7 - Prob. 26E Ch. 1.7 - Prob. 27E Ch. 1.7 - Prob. 28E Ch. 1.7 - Prob. 29E Ch. 1.7 - Prob. 30E Ch. 1.7 - Prob. 31E Ch. 1.7 - Prob. 32E Ch. 1.7 - Prob. 33E Ch. 1.7 - Prob. 34E Ch. 1.7 - Prob. 35E Ch. 1.7 - Prob. 36E Ch. 1.7 - Prob. 37E Ch. 1.7 - Prob. 38E Ch. 1.7 - Prob. 39E Ch. 1.7 - Prob. 40E Ch. 1.7 - Prob. 41E Ch. 1.7 - Prob. 42E Ch. 1.7 - Prob. 43E Ch. 1.7 - Prob. 44E Ch. 1.7 - Prob. 45E Ch. 1.7 - Prob. 46E Ch. 1 - Prob. 1RQ Ch. 1 - Prob. 2RQ Ch. 1 - Prob. 3RQ Ch. 1 - Prob. 4RQ Ch. 1 - Prob. 5RQ Ch. 1 - Prob. 6RQ Ch. 1 - Prob. 7RQ Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Prob. 10RQ Ch. 1 - Prob. 11RQ Ch. 1 - Prob. 12RQ Ch. 1 - (a) What is a function? 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Whether the function f + g is even if f and g are even; whether the function f + g is odd if f and g are odd; whether the function f + g is even or odd if f is even and g is odd.

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Answer to Problem 73E

Explanation of Solution

Chapter 1 Solutions