Problem 1 - A Book of Abstract Algebra

Question

Determine whether each of the following functions is or is not \((a)\) injective, and \((b)\) surjective. \(f: \mathbb{R} \rightarrow(0, \infty)\), defined by \(f(x)=e^{x}\)

Step-by-Step Solution

Answer

The function is both injective and surjective.

Step 1 - Determine Injectivity

To determine if the function is injective, we need to check if every element of the codomain is mapped by at most one element of the domain. For any two elements \(x_1, x_2 \in \mathbb{R}\), consider the equation \(f(x_1) = f(x_2)\). This translates to \(e^{x_1} = e^{x_2}\). Since the exponential function \(e^{x}\) is known to be one-to-one, it implies that \(x_1 = x_2\). Therefore, \(f(x) = e^x\) is injective.

Step 2 - Check Surjectivity

To check if the function is surjective, we need to verify whether every element in the codomain \((0, \, \infty)\) has a preimage in the domain \(\mathbb{R}\). For any \(y \in (0, \infty)\), there exists an \(x \in \mathbb{R}\) such that \(f(x) = y\). We can find \(x\) by taking the natural logarithm: \(x = \ln(y)\). Since \(\ln(y)\) is defined for all \(y > 0\), \(f(x) = e^x\) is surjective from \(\mathbb{R}\) to \((0, \infty)\).

Key Concepts

Exponential Function

The exponential function is a fundamental mathematical concept characterized by the expression \(e^x\), where \(e\) is the base of the natural logarithm, approximately equal to 2.71828. This function has several key properties that make it uniquely interesting. It maps any real number \(x\) to a positive real number, specifically within the range (0, ∞).
What makes the exponential function especially noteworthy is how it grows. For each unit increase in \(x\), the function's output increases multiplicatorily, meaning it grows very rapidly. This property is used extensively in fields such as finance, computer science, and the natural sciences to model exponential growth or decay.
One significant characteristic of \(e^x\) is that it never crosses the x-axis, it only approaches it asymptotically. This means that \(f(x) = e^x\) is always positive, no matter the value of \(x\). By understanding \(e^x\), we gain insight into dynamic systems and natural exponential phenomena.

Injectivity

Injectivity, or being a one-to-one function, is an important concept that tells us how elements from the domain map to elements in the codomain. A function \(f: A \rightarrow B\) is injective if each element of \(B\) is mapped by at most one element of \(A\).
Consider the exponential function \(f(x) = e^x\). To determine its injectivity, let us imagine two different elements from the domain, say \(x_1\) and \(x_2\). If \(f(x_1) = f(x_2)\), this translates to \(e^{x_1} = e^{x_2}\).
Thanks to the properties of logarithms, here we find that \(x_1\) must then equal \(x_2\). Hence, the exponential function \(f(x) = e^x\) is indeed injective. This one-to-one nature guarantees no two different input values will lead to the same output, making this function a valuable tool in scenarios where such unique mapping is required.

Surjectivity

Surjectivity is another core concept that completes our understanding of function mappings. For a function to be surjective (or onto), every element in the codomain must have at least one pre-image in the domain. In simpler terms, the function should cover the entire codomain.
For the exponential function \(f(x) = e^x\), the codomain is set to \((0, \, \infty)\). To verify if it is surjective, we need to show that for any number \(y\) in this range, there exists some real number \(x\) such that \(f(x) = y\). We achieve this by using the natural logarithm, where \(x = \ln(y)\). This calculation always works because \(\ln(y)\) is defined for all positive \(y\).
Therefore, every positive real number \(y\) finds a corresponding value of \(x\) in the domain, affirming that the exponential function is surjective from \(\mathbb{R}\) to \((0, \, \infty)\). Understanding surjectivity helps us ensure that all possible outcomes in our target range are accessible, a vital property in many mathematical applications.