The graph of f is given. (a) Why is f one-to-one? (b) What are the domain and range of f^-1 ? (c

step 1 A function is going to be given by table of values, a graph, a formula, or a verbal description,

The graph of f is given. (a) Why is f one-to-one? (b) What...

Key Concepts

Graphical Estimation of Inverse Values

Graphical estimation involves reading the inverse function values from the graph of the original function. To estimate f?¹(a), one locates the point on the graph of f where the output equals a and then determines the corresponding input value. This visual method is especially useful when dealing with functions for which precise algebraic manipulation is complex or when an exact analytical solution is not feasible.

Domain and Range Swapping in Inverses

The domain and range of an inverse function are switched relative to the original function. Specifically, the domain of f?¹ is the range of f, and the range of f?¹ is the domain of f. This concept is central to understanding how the characteristics of a function translate when forming its inverse.

Inverse Functions

An inverse function reverses the mapping of the original function. If f takes an input x and produces an output y (i.e., f(x) = y), then its inverse function f?¹ takes y as the input and returns x (i.e., f?¹(y) = x). This relation is fundamental in understanding how functions can be systematically inverted, provided they are one-to-one.

One-to-One Functions and the Horizontal Line Test

A function is considered one-to-one (injective) if every horizontal line intersects its graph at most once. This means that no two different input values produce the same output. The horizontal line test is a visual way to determine this property from the graph, ensuring that the function has a unique inverse.

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