Question

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

Question: Determine whether each of the following statements is true or not and provide 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 \(x\). Answer: a. True b. False c. True d. False e. False f. True g. True h. False i. True

Step-by-step solution

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

Given the function \(f(x) = 2x - 38\), we see that it is a linear function, and as a result, the range of linear functions is always all real numbers. This statement is true.

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

This statement is false. The given relation is a function, as each value of \(x\) uniquely determines the value of \(y\). While it's true that \(y=2\) for both \(x=-1\) and \(x=1\), it doesn't imply that the relation is not a function. For a relation to not be a function, we would need a single \(x\) value to be associated with more than one \(y\) value, which doesn't happen here.

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

Given \(f(x) = x^{-1}\), let's find the value of \(f\left( \frac{1}{x} \right)\) and \(f(x)\): \(f\left(\frac{1}{x}\right) = \left(\frac{1}{x}\right)^{-1} = x\). Now we'll find \(\frac{1}{f(x)} = \frac{1}{x^{-1}} = x\). Since both expressions are equal, this statement is true.

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

This statement is false, as it's not valid for all functions. A counterexample: Let \(f(x) = -x\). Then, \(f(f(x)) = f(-x) = -(-x) = x\) while \((f(x))^2 = (-x)^2 = x^2\). Since the expressions are not equal, this statement is false.

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

This statement is false. This is only true if \(f\) and \(g\) commute, which doesn't always happen. A counterexample: Let \(f(x) = x + 1\) and \(g(x) = x^2\). Then, \(f(g(x)) = f(x^2) = x^2 + 1\), while \(g(f(x)) = g(x + 1) = (x + 1)^2\). Since the expressions are not equal, this statement is false.

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

This statement is true by definition of function composition. The composition of \(f\) and \(g\), denoted as \((f\circ g)(x)\), is equal to \(f(g(x))\).

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.

To prove this statement, we must show that \((cf(ax)) = cf(-ax)\). Assume that \(f(x)\) is an even function, which means that \(f(x) = f(-x)\). Then, \(cf(ax) = c f(-a(-x)) = c f(-ax)\). Since the expression holds true, this 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.

This statement is false. Suppose \(f(x)\) is an odd function, that means \(f(x) = -f(-x)\). Now, let's check if \(f(x) + d\) is odd: \((f(x)+d) + (f(-x) + d) = f(x) + d - f(-x) - d = f(x) - f(-x)\). Since this expression is not equal to zero, the given function is not odd. Therefore, this statement is false.

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

Suppose \(f\) is both even and odd. As an even function, we have \(f(x) = f(-x)\), and as an odd function, we have \(f(x) = -f(-x)\). Combining these relations, we get \(f(x) = -f(-x) = f(x)\). Therefore, \(f(x) - f(x) = 0\), which implies that \(f(x) = 0\) for all \(x\). This statement is true.

Key concepts

Linear Functions

Linear functions are one of the simplest types of functions in calculus and algebra. A linear function can be defined by the equation: \[ f(x) = mx + b \]where \(m\) is the slope, and \(b\) is the y-intercept. Linear functions are so named because their graphs produce straight lines. There are several key characteristics of linear functions:

• They are defined for all real numbers, and their range is also all real numbers.
• The slope \(m\) dictates whether the line is increasing, decreasing, or horizontal. A positive slope indicates that the line rises as \(x\) increases, a negative slope falls, and a slope of zero means the line is horizontal.
• The y-intercept \(b\) represents the point where the line crosses the y-axis.

It's important to understand that the simplicity and predictability of linear functions make them very useful in various applications, such as modeling real-world situations like speed versus time graphs.

Function Composition

Function composition involves creating a new function by applying one function to the result of another. If \(f\) and \(g\) are two functions, the composition \((f \circ g)(x)\) is defined by:\[ (f \circ g)(x) = f(g(x)) \]Function composition is a fundamental concept in calculus and can be thought of as plugging one function into another. A few important points about function composition:

• Function composition is not necessarily commutative, which typically means \(f(g(x)) eq g(f(x))\).
• The order of operations is essential in composition; changing the order generally changes the function.
• It allows for flexible application in solving complex mathematical problems by breaking them into simpler parts.

Understanding function composition can greatly enhance problem-solving skills, especially in more advanced mathematical contexts like calculus and in functional programming.

Even and Odd Functions

Even and odd functions refer to symmetry in the graph of a function. An even function is symmetric about the y-axis, meaning:\[ f(x) = f(-x) \]On the other hand, an odd function exhibits symmetry about the origin, defined by:\[ f(x) = -f(-x) \]To determine if a function is even, you can replace \(x\) with \(-x\) and see if the function remains unchanged. For an odd function, doing the same should yield the negative of the original function. Here are some characteristics:

• Even functions have graphs that fold symmetrically along the y-axis.
• Odd functions possess a rotational symmetry around the origin at 180 degrees.
• Understanding even and odd functions can be useful in integrating functions and graphing because of their symmetry properties.
• Functions can also be neither even nor odd or could be both if they are zero functions.

These properties can simplify many calculations and help in understanding the behavior of function graphs.

Counterexamples

A counterexample is employed in mathematics to show that a certain statement or conjecture is false. Providing a counterexample involves finding an instance where the statement doesn't hold:

• Counterexamples are crucial in disproving false claims or misconceptions in mathematical proofs or theories.
• It usually involves just a single example, yet that one example is sufficient to negate a universal claim.
• Identifying effective counterexamples requires a good understanding of the underlying concepts and predicates being examined.

For instance, the statement “In general, \(f(g(x)) = g(f(x))\)” can be disproved by a counterexample using simple functions. If \(f(x) = x + 1\) and \(g(x) = x^2\), then \(f(g(x))\) and \(g(f(x))\) are indeed different.
Counterexamples play a vital role in learning and clarifying mathematical concepts, ensuring that understandings and theories are robust and accurate.