Is the Statement True? Analyzing Mathematical Functions

When dealing with mathematical functions, it is crucial to have a deep understanding of their properties and behaviors. In this article, we will explore the truth or falsehood of two specific statements involving the hyperbolic tangent function (tanh x) and the reciprocal exponential function (1/e^x).

Understanding the Hyperbolic Tangent Function (tan hx)

The hyperbolic tangent function, denoted as tan hx, is defined for all real numbers x. Its mathematical representation is given by the formula:

tan hx (e^x - e^{-x}) / (e^x e^{-x})

One important property of the tan hx function is that it maps any real number to the interval (-1, 1). This means that for any value of x, the output of tan hx will never be outside this range. This makes it a valuable tool in various mathematical and computational applications.

Counterexample for the Hyperbolic Tangent Function

The statement "No. The function tan hx is a counterexample" suggests that tan hx serves as a counterexample to some property or statement. Given the definition and properties of tan hx, this statement may be referring to a specific claim that does not hold for this function. Let's delve deeper into why this might be the case.

Is the Argument True or False?

The function tan hx is a bijective (one-to-one and onto) function from the set of real numbers to the open interval (-1, 1). Since it is bijective, every value in the codomain (the range of (-1, 1)) is mapped from exactly one value in the domain. Therefore, tan hx cannot map any number to a negative number less than -1 or to a number greater than 1. This is a fundamental property of the tan hx function.

Counterargument with tan hx

Consider the statement: "No. tan hx is a counterexample." This statement is true because the tan hx function does not map any number to a value outside the interval (-1, 1). Hence, any claim that tan hx would map a number to a negative number less than -1 or to a number greater than 1 is false. This is the core of the counterexample argument.

For example, if we take any real number x, the value of tan hx will always be between -1 and 1. If the statement were to suggest that tan hx maps to a number greater than 1 or less than -1 for some x, this would be contradictory to the definition and properties of the tan hx function.

Understanding the Reciprocal Exponential Function

Next, let's consider the statement: "This is not true. For example, f(x) 1/e^x doesn’t map any number to a negative number or to any number greater than 1."

The function f(x) 1/e^x can be rewritten as f(x) e^{-x}. This is an exponential decay function and its properties are quite different from the tan hx function.

Properties of the Reciprocal Exponential Function

The function f(x) e^{-x} has the following key properties:

Domain: The domain of f(x) e^{-x} is all real numbers, i.e., (-∞, ∞). Range: The range of f(x) e^{-x} is the positive real numbers, i.e., (0, ∞). Behavior: The function f(x) e^{-x} is always positive and never crosses the x-axis. It approaches 0 as x goes to infinity and approaches infinity as x goes to negative infinity.

Given these properties, it is clear that f(x) e^{-x} never maps any number to a negative number. Moreover, since the range of this function is (0, ∞), it also never maps any number to a value outside this positive range. Therefore, it indeed does not map any number to a negative number or to any number greater than 1.

Conclusion

To summarize, the statement "For example, f(x) 1/e^x doesn’t map any number to a negative number or to any number greater than 1" is correct. The function 1/e^x is a well-defined, positive, and decreasing function that maps all real numbers to the positive real numbers.

Understanding these functions and their properties is essential for various mathematical and computational applications. Whether it is the hyperbolic tangent function or the reciprocal exponential function, knowledge of their domains, ranges, and behaviors helps in making accurate predictions and developing robust mathematical models.