Lecture Notes on Inverse Functions

These flashcards cover key concepts from the lecture on inverse functions, including definitions, properties of functions, and asymptotic behavior.

One-to-one function

A function is one-to-one if it never takes the same value twice, meaning that f(x1) ≠ f(x2) whenever x1 ≠ x2.

Horizontal line test

A graph passes the horizontal line test if each horizontal line cuts the graph at most once.

Inverse function, f^{-1}(y)

If f is a one-to-one function with domain A and range B, the inverse function f^{-1}(y) is defined by the rule f^{-1}(y) = x if f(x) = y.

Graph of a one-to-one function

If f is a one-to-one function, no two points (x1, y1) and (x2, y2) have the same y-values.

Finding an inverse formula

To find the formula for f^{-1}(x), solve y = f(x) for x in terms of y and then swap x and y.

Vertical asymptote

A vertical asymptote occurs where the function does not exist at x = a; it represents x-values not allowed.

Horizontal asymptote

A horizontal asymptote occurs when the limit of f(x) as x approaches ±∞ equals a constant c.

Diagonal (slant) asymptote

A diagonal asymptote occurs when the degree of the numerator is one higher than the degree of the denominator.

Natural logarithm function, ln(x)

The natural logarithm function is the inverse function of the exponential function e^x.

L'Hôpital's Rule

A method used to find the limit of functions that result in indeterminate forms (0/0 or ∞/∞) by taking derivatives.