(a) To determine To classify: The function f ( x ) = log 2 x as one of the type of functions. Answer The function f ( x ) = log 2 x is a logarithmic function. Explanation Reason: The function is of the form f ( x ) = log b x , where b is a positive constant is said to be a logarithmic function. It is a reciprocal of an exponential function. (b) To determine To classify: The function g ( x ) = x 4 as one of the type of functions. Answer The function g ( x ) = x 4 is a root function. Explanation Reason: The function is of the form f ( x ) = x n , is said to be a root function. Note that the expression inside the root is always positive. (c) To determine To classify: The function h ( x ) = 2 x 3 1 − x 2 as one of the type of functions. Answer The function h ( x ) = 2 x 3 1 − x 2 is a rational function. Explanation Reason: The function is of the form f ( x ) = p ( x ) q ( x ) , where p ( x ) and q ( x ) are the polynomials and q ( x ) ≠ 0 is said to be a rational function. Here, p ( x ) = 2 x 3 and q ( x ) = 1 − x 2 . Thus, the function h ( x ) is a rational function. (d) To determine To classify: The function u ( t ) = 1 − 1.1 t + 2.54 t 2 as one of the type of functions. Answer The function u ( t ) = 1 − 1.1 t + 2.54 t 2 is a polynomial function with degree 2. Explanation Reason: Rewrite the given function as u ( t ) = 2.54 t 2 − 1.1 t + 1 . The polynomial function is of the form p ( x ) = a n x n + a n − 1 x n − 1 + ... + a 1 x + a 0 , where n is a positive integer and a 0 , a 1 , ... , a n are constants. Thus, the given function is a polynomial function with degree 2. Therefore, the function is a quadratic function. (e) To determine To classify: The function v ( t ) = 5 t as one of the type of functions. Answer The function v ( t ) = 5 t is an exponential function. Explanation Reason: An exponential function is of the form f ( x ) = b x , where b is a positive constant and x is an exponent. Here, b = 5 and x = t . Thus, the given function is said to be an exponential function. (f) To determine To classify: The function w ( θ ) = sin θ cos 2 θ as one of the type of functions. Answer The function w ( θ ) = sin θ cos 2 θ is a trigonometric function. Explanation Reason: The function involves with the trigonometric functions such as sin θ and cos θ . Thus, the given function is said to bea trigonometric function.

Single Variable Calculus: Concepts and Contexts, Enhanced Edition

4th Edition

ISBN: 9781337687805

Author: James Stewart

Publisher: Cengage Learning

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Textbook Question

Chapter 1.2, Problem 1E

Classify each function as a power function, root function, polynomial (state its degree), rational function, algebraic function, trigonometric function, exponential function, or logarithmic function.

1. (a) f ( x ) = log 2 x

(b) g ( x ) = x 4

(d) u ( t ) = 1 − 1.1 t + 2.54 t 2

(e) v ( t ) = 5 t

(f) w ( θ ) = sin θ c o s 2 θ

(a)

Expert Solution

To determine

To classify: The function f(x)=log2x as one of the type of functions.

Answer to Problem 1E

The function f(x)=log2x is a logarithmic function.

Explanation of Solution

Reason:

The function is of the form f(x)=logbx , where b is a positive constant is said to be a logarithmic function. It is a reciprocal of an exponential function.

(b)

Expert Solution

To determine

To classify: The function g(x)=x4 as one of the type of functions.

Answer to Problem 1E

The function g(x)=x4 is a root function.

Explanation of Solution

Reason:

The function is of the form f(x)=xn , is said to be a root function. Note that the expression inside the root is always positive.

(c)

Expert Solution

To determine

To classify: The function h(x)=2x31−x2 as one of the type of functions.

Answer to Problem 1E

The function h(x)=2x31−x2 is a rational function.

Explanation of Solution

Reason:

The function is of the form f(x)=p(x)q(x) , where p(x) and q(x) are the polynomials and q(x)≠0 is said to be a rational function. Here, p(x)=2x3 and q(x)=1−x2 . Thus, the function h(x) is a rational function.

(d)

Expert Solution

To determine

To classify: The function u(t)=1−1.1t+2.54t2 as one of the type of functions.

Answer to Problem 1E

The function u(t)=1−1.1t+2.54t2 is a polynomial function with degree 2.

Explanation of Solution

Reason:

Rewrite the given function as u(t)=2.54t2−1.1t+1 .

The polynomial function is of the form p(x)=anxn+an−1xn−1+...+a1x+a0 , where n is a positive integer and a0,a1,...,an are constants. Thus, the given function is a polynomial function with degree 2. Therefore, the function is a quadratic function.

(e)

Expert Solution

To determine

To classify: The function v(t)=5t as one of the type of functions.

Answer to Problem 1E

The function v(t)=5t is an exponential function.

Explanation of Solution

Reason:

An exponential function is of the form f(x)=bx , where b is a positive constant and x is an exponent. Here, b=5 and x=t . Thus, the given function is said to be an exponential function.

(f)

Expert Solution

To determine

To classify: The function w(θ)=sinθcos2θ as one of the type of functions.

Answer to Problem 1E

The function w(θ)=sinθcos2θ is a trigonometric function.

Explanation of Solution

Reason:

The function involves with the trigonometric functions such as sinθ and cosθ . Thus, the given function is said to bea trigonometric function.

Chapter 1.1, Problem 74E

Chapter 1.2, Problem 2E

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. 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(b) How can you...Ch. 1.6 - Prob. 2E Ch. 1.6 - Prob. 3E Ch. 1.6 - Prob. 4E Ch. 1.6 - A function is given by a table of values, a graph,...Ch. 1.6 - A function is given by a table of values, a graph,...Ch. 1.6 - Prob. 7E Ch. 1.6 - A function is given by a table of values, a graph,...Ch. 1.6 - Prob. 9E Ch. 1.6 - Prob. 10E Ch. 1.6 - Prob. 11E Ch. 1.6 - Prob. 12E Ch. 1.6 - Prob. 13E Ch. 1.6 - Prob. 14E Ch. 1.6 - Assume that f is a one-to-one function. (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? What are its domain and...Ch. 1 - Discuss four ways of representing a function....Ch. 1 - (a) What is an even function? How can you tell if...Ch. 1 - Prob. 4RCC Ch. 1 - Prob. 5RCC Ch. 1 - Prob. 6RCC Ch. 1 - Prob. 7RCC Ch. 1 - Prob. 8RCC Ch. 1 - Suppose that f has domain A and g has domain B....Ch. 1 - Prob. 10RCC Ch. 1 - Prob. 11RCC Ch. 1 - Prob. 12RCC Ch. 1 - Prob. 13RCC Ch. 1 - Let f be the function whose graph is given. (a)...Ch. 1 - Prob. 2RE Ch. 1 - Prob. 3RE Ch. 1 - Prob. 4RE Ch. 1 - Prob. 5RE Ch. 1 - Prob. 6RE Ch. 1 - Prob. 7RE Ch. 1 - Prob. 8RE Ch. 1 - Suppose that the graph of .f is given. Describe...Ch. 1 - Prob. 10RE Ch. 1 - Prob. 11RE Ch. 1 - Prob. 12RE Ch. 1 - Prob. 13RE Ch. 1 - Prob. 14RE Ch. 1 - Prob. 15RE Ch. 1 - Prob. 16RE Ch. 1 - Prob. 17RE Ch. 1 - Prob. 18RE Ch. 1 - Prob. 19RE Ch. 1 - Prob. 20RE Ch. 1 - Prob. 22RE Ch. 1 - Prob. 23RE Ch. 1 - Prob. 24RE Ch. 1 - Prob. 25RE Ch. 1 - Prob. 26RE Ch. 1 - The half-life of palladium-100, 100Pd, is four...Ch. 1 - The population of a certain species in a limited...Ch. 1 - Prob. 29RE Ch. 1 - Prob. 30RE Ch. 1 - Prob. 31RE Ch. 1 - Prob. 32RE Ch. 1 - Prob. 33RE Ch. 1 - Prob. 34RE Ch. 1 - One of the legs of a right triangle has length 4...Ch. 1 - Prob. 2P Ch. 1 - Prob. 3P Ch. 1 - Prob. 4P Ch. 1 - Prob. 5P Ch. 1 - Prob. 6P Ch. 1 - Prob. 7P Ch. 1 - Prob. 8P Ch. 1 - Prob. 9P Ch. 1 - Prob. 10P Ch. 1 - Prob. 11P Ch. 1 - Prob. 12P Ch. 1 - Prob. 13P Ch. 1 - Prob. 14P Ch. 1 - Prob. 15P Ch. 1 - Prob. 16P Ch. 1 - Prob. 17P Ch. 1 - Prove that 1 + 3 + 5 + + (2n l ) = n2.Ch. 1 - Prob. 19P Ch. 1 - Prob. 20P

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