91Ó°ÊÓ

An injective function, or one-to-one function, is a function in which different elements in the domain map to different elements in the codomain. This means that if \(f(a) = f(b)\), then \(a = b\). To show that a function \(f\) is injective, you can take any two elements \(a\) and \(b\) in the domain. Assume that \(f(a) = f(b)\). Through algebraic manipulation or other methods, you should be able to conclude that \(a = b\).

For example, let's look at the function \(f(x) = e^x\) over the real numbers. Suppose \(e^a = e^b\). If we take the natural logarithm of both sides, we get \(a = b\). Therefore, the function \(f(x) = e^x\) is injective on the set of real numbers, because different inputs lead to different outputs.

A surjective function, or onto function, is a function where every element in the codomain is mapped to by at least one element in the domain. In other words, if \(f: A \to B\) is surjective, for every \(b\) in \(B\), there exists an \(a\) in \(A\) such that \(f(a) = b\).

Consider the function \(f(x) = e^x\) again, but this time we look at its surjectivity over the real numbers \(\mathbb{R}\). The exponential function \(e^x\) maps any real number \(x\) to a positive real number. This means that not all real numbers are covered since negative numbers are left out. Because of this, \(f(x) = e^x\) is not surjective when mapping from \(\mathbb{R}\) to \(\mathbb{R}\). However, if we restrict the codomain to positive real numbers, \(\mathbb{R}^{+}\), every positive number will have an \(x\) in the domain such that \(e^x = y\). This restriction makes the function surjective.

An exponential function is a mathematical function of the form \(f(x) = a^{bx}\), where \(a > 0\) is a constant base and \(b\) is the exponent. Exponential functions are widely used in many fields such as biology, chemistry, and finance to model growth processes.

One common exponential function is \(f(x) = e^x\), where \(e\) is the Euler's number, approximately equal to 2.71828. This function has some unique properties:

• It is always positive regardless of \(x\).
• It continuously increases as \(x\) increases.
• The slope of \(e^x\) at any point is equal to its value at that point.

Despite these properties, \(f(x) = e^x\) is not invertible when mapping from all real numbers to all real numbers because it does not cover negative real numbers. However, if we restrict the codomain to positive real numbers, it becomes invertible. This means there exists a unique inverse function \(f^{-1}(x)\), which happens to be the natural logarithm function \(\ln(x)\). This inverse relationship is crucial in solving equations involving exponentials.