91Ó°ÊÓ

Rational functions are a special kind of algebraic function. They are defined as the ratio of two polynomials. In the function \( f(x) = \frac{x-2}{x+2} \), the numerator is the polynomial \( x - 2 \) and the denominator is \( x + 2 \). These types of functions have unique properties worth exploring.

One important characteristic is their **domain**, which consists of all real numbers except those that make the denominator zero. This happens because division by zero is undefined. For \( f(x) = \frac{x-2}{x+2} \), the denominator becomes zero at \( x = -2 \). Hence, \( x = -2 \) is excluded from the domain of \( f \).

Furthermore, rational functions can have vertical and horizontal asymptotes. They are lines that the graph of the function approaches but never touches. The vertical asymptote of our function is at \( x = -2 \), while the horizontal asymptote can be determined by observing the degrees of the polynomials. Since both the numerator and the denominator are first-degree polynomials, the horizontal asymptote is \( y = 1 \), the ratio of their leading coefficients.

Function composition involves applying one function to the result of another function. When dealing with inverse functions, composition is a test to verify if two functions are indeed inverses. Specifically, if \( f(x) \) and \( f^{-1}(x) \) are inverses, then \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).

To compose two functions, insert the output from one function into the input of the other. For example, with the functions \( f(x) = \frac{x-2}{x+2} \) and \( f^{-1}(x) = \frac{-2(x+1)}{x-1} \):

• Compute \( f(f^{-1}(x)) = \frac{x-2}{x+2} \), replace \( x \) with \( f^{-1}(x) \) and simplify.
• Likewise, verify \( f^{-1}(f(x)) = x \) by substituting \( f(x) \) into \( f^{-1} \).

Ensuring both compositions return \( x \) confirms the inverse relationship.

Solving equations is an essential skill used to manipulate mathematical statements to isolate a variable. In finding inverse functions, solving equations helps in rearranging and isolating variables.

The process starts by setting the function equal to \( y \), as seen with \( y = \frac{x-2}{x+2} \). To **solve for \( x \)**, follow these steps:

• Eliminate fractions by multiplying each term by the denominator. This helps clear out division issues.
• Rearrange the equation to isolate terms involving \( x \). Always bring terms containing \( x \) to one side of the equation.
• Factor out \( x \) if it is present in multiple terms, to consolidate it.
• The final step involves solving for \( x \), wherein you divide to isolate \( x \).

Switch variables to express the inverse, transforming the solved \( x \) form back into a function of \( y \). This inverse becomes \( f^{-1}(x) \). This systematic approach ensures that the provided solution verifies the equation and confirms that the inverse relation is correctly derived.