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The domain of a function refers to the set of all possible input values (usually represented as "x") that the function can accept without leading to any undefined scenarios. When we talk about a rational function like \( f(x) = \frac{1}{x} \), we encounter a classic example of a domain restriction.

Because division by zero is undefined, the domain for this function includes all real numbers except zero. In mathematical notation, this is expressed as:

• Domain of \( f(x) = \frac{1}{x} \): all real numbers \( x \), except \( x eq 0 \).

This means you can input any number into \( f(x) = \frac{1}{x} \) except zero, and you'll get a valid output. In any function problem, always start by identifying its domain, as it sets the stage for understanding more complex properties like one-to-one behavior.

An injective function, also known as a one-to-one function, is special because each element in its domain maps to a unique output in its range. For a function \( f(x) \) to be injective, if you take any two different inputs, having the same output must necessarily imply that the inputs were identical to begin with.

Mathematically, for any distinct \( a \) and \( b \) in the domain:

• If \( f(a) = f(b) \), then it must be true that \( a = b \).

In our specific example with \( f(x) = \frac{1}{x} \), checking this injective property involves setting \( f(a) = f(b) \) and showing that this leads to \( a = b \). Multiplying through by \( a \cdot b \) to clear the fractions leads us straight to this conclusion. Once this is confirmed for all values in the domain, we can confidently say \( f(x) = \frac{1}{x} \) is an injective function.

Rational functions are functions defined by the ratio of two polynomials. The standard form is \( f(x) = \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials.

The function \( f(x) = \frac{1}{x} \) is a simple example of a rational function where our \( P(x) = 1 \) and \( Q(x) = x \). In rational functions, the denominator \( Q(x) \) must not be zero, as this would make the function undefined. This requirement is also what determines the domain of rational functions, excluding any x-values that make the denominator zero.
The fascinating aspect of rational functions is their behavior in terms of asymptotes: lines that the graph approaches but never quite touches.

• Vertical asymptotes occur at points where the function is undefined.
• Horizontal or slant asymptotes often depend on the degrees of the polynomials.

Understanding and analyzing these features is crucial for graphing and fully grasping rational functions.