Problem 61 Determine whether the following ... [FREE SOLUTION]

Chapter 1: Problem 61

Determine whether the following statements are true and give an explanation or counterexample. a. The range of \(f(x)=2 x-38\) is all real numbers. b. The relation \(y=x^{6}+1\) is not a function because \(y=2\) for both \(x=-1\) and \(x=1\) c. If \(f(x)=x^{-1}\), then \(f(1 / x)=1 / f(x)\) d. In general, \(f(f(x))=(f(x))^{2}\) e. In general, \(f(g(x))=g(f(x))\) f. By definition, \(f(g(x))=(f \circ g)(x)\) g. If \(f(x)\) is an even function, then \(c f(a x)\) is an even function, where \(a\) and \(c\) are nonzero real numbers. h. If \(f(x)\) is an odd function, then \(f(x)+d\) is an odd function, where \(d\) is a nonzero real number. i. If \(f\) is both even and odd, then \(f(x)=0\) for all \(x\)

Short Answer

Expert verified

a. True. The range of \(f(x)=2x-38\) is all real numbers because as \(x\) goes to positive or negative infinity, the function values also go to positive or negative infinity respectively. b. False. The relation \(y=x^6+1\) is a function because each input (value of \(x\)) corresponds to exactly one output (value of \(y\)). The given statement discusses the points with the same \(y\) values, not the same \(x\) values, so it is not a proper reasoning for it to be not a function. c. True. If \(f(x)=x^{-1}\), then \(f(1 / x)=1 / f(x)\). d. False. In general, \(f(f(x))\neq(f(x))^2\). An example is the function \(f(x) = x+1\). e. False. In general, \(f(g(x)) \neq g(f(x))\). An example is the functions \(f(x)=x^2\) and \(g(x)=x+1\). f. True. By definition, \(f(g(x))=(f \circ g)(x)\). g. True. If \(f(x)\) is an even function, then \(cf(ax)\) is an even function where \(a\) and \(c\) are nonzero real numbers. h. False. If \(f(x)\) is an odd function, then \(f(x)+d\) is not necessarily an odd function where \(d\) is a nonzero real number. i. True. If \(f\) is both even and odd, then \(f(x)=0\) for all \(x\).

Step by step solution

a. The range of \(f(x)=2x-38\) is all real numbers.

Since \(f(x) = 2x - 38\), it is a linear function with a slope of 2 and a \(y\)-intercept of -38. As \(x\) goes to positive or negative infinity, the function values also go to positive or negative infinity, respectively. Therefore, the range of \(f(x)=2x-38\) is all real numbers. Thus, the statement is true.

b. The relation \(y=x^6+1\) is not a function because \(y=2\) for both \(x=-1\) and \(x=1\).

A relation is a function if each input (value of \(x\)) corresponds to exactly one output (value of \(y\)). The given statement discusses the points with the same \(y\) values, not the same \(x\) values, so it is not a proper reasoning for it to be not a function. Furthermore, since the equation \(y=x^6+1\) satisfies the criterion of function definition, this statement is false.

c. If \(f(x)=x^{-1}\), then \(f(1 / x)=1 / f(x)\).

Let's find \(f(1/x)\): \(f(1/x) = (1/x)^{-1}=x\). Now let's find \(1/f(x)\): \(1/f(x) = 1/(x^{-1}) = x\). Since \(f(1/x)=1/f(x)\), the statement is true.

d. In general, \(f(f(x))=(f(x))^2\).

Consider the counterexample \(f(x) = x+1\). Then, \(f(f(x)) = f(x+1) = (x+1) + 1 = x+2\), \((f(x))^2 = (x+1)^2 = x^2 + 2x + 1\). In this case, \(f(f(x)) \neq (f(x))^2\). Thus, the statement is false.

e. In general, \(f(g(x))=g(f(x))\).

Consider the counterexample \(f(x)=x^2\) and \(g(x)=x+1\). Then, \(f(g(x)) = f(x+1) = (x+1)^2\), \(g(f(x)) = g(x^2) = x^2+1\). In this case, \(f(g(x)) \neq g(f(x))\). Thus, the statement is false.

f. By definition, \(f(g(x))=(f \circ g)(x)\).

The statement is true, as the composition of functions \(f\) and \(g\) is defined as \((f \circ g)(x)=f(g(x))\).

g. If \(f(x)\) is an even function, then \(cf(ax)\) is an even function, where \(a\) and \(c\) are nonzero real numbers.

Suppose \(f(x)\) is an even function, which means \(f(-x)=f(x)\). Now, let \(g(x)=cf(ax)\), where \(a\) and \(c\) are nonzero real numbers. \(g(-x) = cf(-ax) = c\cdot f(ax)\) (since \(f(x)\) is even), \(g(x) = cf(ax)\). Since \(g(-x)=g(x)\), the statement is true.

h. If \(f(x)\) is an odd function, then \(f(x)+d\) is an odd function, where \(d\) is a nonzero real number.

Suppose \(f(x)\) is an odd function, which means \(f(-x)=-f(x)\). Now, let \(g(x)=f(x)+d\), where \(d\) is a nonzero real number. \(g(-x) = f(-x)+d = -f(x)+d\) (since \(f(x)\) is odd), \(g(x) = f(x)+d\). Since \(g(-x) \neq -g(x)\) in this case, the statement is false.

i. If \(f\) is both even and odd, then \(f(x)=0\) for all \(x\).

If \(f\) is even, then \(f(-x)=f(x)\). If \(f\) is odd, then \(f(-x)=-f(x)\). So, if \(f\) is both even and odd, we have \(f(x)=f(-x)=-f(x)\), which implies \(f(x)=-f(x)\). The only function that satisfies this condition for all values of \(x\) is \(f(x)=0\). Thus, the statement is true.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Function Range

Understanding the range of a linear function is crucial when analyzing its behavior. In calculus, the range of a linear function, such as \(f(x)=2x-38\), refers to the set of all possible output values the function can produce. Since linear functions have the general form \(f(x)=mx+b\) where \(m\) and \(b\) are constants representing the slope and y-intercept respectively, they create straight lines when graphed. In the case of \(f(x)=2x-38\), no matter what real number you substitute for \(x\), the output will be a real number. This is because the slope \(m=2\) ensures that as \(x\) increases or decreases, the output value will adjust accordingly without any restrictions. Hence, the range of this function is indeed all real numbers, just as the step-by-step solution concluded.

Definition of a Function

A function is a foundational concept in calculus and higher mathematics. In its essence, a function describes a special relationship between two sets: the domain and the range. For each element in the domain, a function assigns precisely one element in the range. This is sometimes referred to as the 'unique output' rule. The exercise incorrectly challenged the function status of \(y=x^{6}+1\), mistaking equal outputs for different inputs as a violation of the function definition. However, the true function test concerns unique inputs rather than unique outputs. The given relation passes this test since every input value of \(x\) indeed corresponds to one output value of \(y\), confirming that it is, without a doubt, a function.

Composition of Functions

The composition of functions is a key operation in calculus, allowing us to combine two functions into a single function by using the output of one function as the input of another. This process is denoted by \((f \(\circ\) g)(x)\), which means '\(f\) composed with \(g\).' The correct interpretation, as was shown in the step-by-step solution, is that \((f \(\circ\) g)(x) = f(g(x))\). Although it is tempting to assume certain properties about composition, such as \(f(f(x))=(f(x))^2\) or \(f(g(x))=g(f(x))\), these are generally not true for all functions. Specific examples used in the solution demonstrate that these properties may hold under certain conditions, but they cannot be applied universally, underscoring the importance of treating compositions on a case-by-case basis.

Properties of Even and Odd Functions

Even and odd functions have distinct properties that can be particularly handy when solving problems involving symmetry. An even function satisfies the characteristic equation \(f(-x) = f(x)\), which implies that its graph is symmetrical with respect to the y-axis. Conversely, an odd function satisfies \(f(-x) = -f(x)\), meaning its graph is symmetrical with respect to the origin. These properties extend beyond the functions themselves to transformations involving constants, as seen in the textbook solutions. When scaling even functions with constants (other than zero), the result remains even. However, adding a constant to an odd function does not preserve its oddness, altering the function's symmetry and often its general behavior. Recognizing these properties aids in predicting the effects of transformations, like scaling and shifting, on a function's graph and functionality.

Function Transformation

Function transformation involves modifying a function's formula in a way that affects its graph. This can include scaling, shifting, reflecting, or stretching the graph of the function. Understanding how these transformations relate to the function's algebraic expression is essential for analyzing and graphing functions. For instance, multiplying a function by a scalar stretches or compresses it vertically, while adding a constant inside the function argument results in a horizontal shift. The exercise reinforced the principle that if a function is even or odd, certain transformations will maintain those properties. However, other transformations, like adding a constant to an odd function, do not preserve the original symmetry, leading to a new function with distinct properties. Through practice, students can gain a deeper comprehension of how alterations to the function's formula impact its graphical representation and intrinsic properties.

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Short Answer

Step by step solution

a. The range of \(f(x)=2x-38\) is all real numbers.

b. The relation \(y=x^6+1\) is not a function because \(y=2\) for both \(x=-1\) and \(x=1\).

c. If \(f(x)=x^{-1}\), then \(f(1 / x)=1 / f(x)\).

d. In general, \(f(f(x))=(f(x))^2\).

e. In general, \(f(g(x))=g(f(x))\).

f. By definition, \(f(g(x))=(f \circ g)(x)\).

g. If \(f(x)\) is an even function, then \(cf(ax)\) is an even function, where \(a\) and \(c\) are nonzero real numbers.

h. If \(f(x)\) is an odd function, then \(f(x)+d\) is an odd function, where \(d\) is a nonzero real number.

i. If \(f\) is both even and odd, then \(f(x)=0\) for all \(x\).

Key Concepts

Linear Function Range

Definition of a Function

Composition of Functions

Properties of Even and Odd Functions

Function Transformation

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