2.8 Inverse functions
What is an inverse function?
What does it mean for a function to be invertible?
If a function, f, and its inverse function, f^{ −1}, are composed, what is the result?
How do the domain and range of a function relate to the domain and range of its inverse function?
What are the two methods for finding the inverse of a function?
Given the function f(x) = 2x + 3, find its inverse function.
The function g(x) = x^3 + 1 is one-to-one. Find its inverse function.
Consider the function f(x) = 2x + 1. Determine the domain and range of its inverse.
Consider the function j(x) = 2x^2 - 3 on the domain [1.5, \infty). Find its inverse function.
A function k(x) has the following table of values: (2,3), (4,1), (3,-2), (1,4). Find the table of values for its inverse function.
The graph of function l(x) passes through the points (1,2), (2,3), (3,4). Sketch the graph of its inverse function.
The graph of a function y = f(x) passes through the points (2,\,3) and(5,\,-1). If the function is invertible, what points must be on the graph of y = f^{-1}(x)?
The function n(x) = 3x - 7 represents the cost in dollars to produce x items. Find the inverse function and explain its practical interpretation in the context of the problem.
A function f is defined as f(x) = x^2 for x \geq 0 and has an inverse function f^{-1}. Given that the graph of f passes through the points (1,\,1) and (2,\,4), what points must be on the graph of? Explain your reasoning.
Consider the function p(x) = (x - 1)(x - 3)^2(x + 2). This function is not invertible in its current domain. Can you think of a real-world situation where we would be interested in the inverse of this function, but only need it to be defined on a restricted domain? What would that domain be?
Consider the function q(x) = 5x^3 - 2x + 1. This function is invertible, but the inverse function is not a simple algebraic function. How could we use a graphing tool or numerical methods to approximate the value of q^{-1}(3)?
The function r(x) = \dfrac{1}{2}x^2 is defined for all real numbers, but it is only invertible if we restrict the domain.
What is the inverse function if we restrict the domain to nonnegative numbers?
What about if we restrict the domain to nonpositive numbers?
What would be the practical implications of these two different choices in a real-world application?
Consider the function h(x) = \sqrt{x - 2}. This function is an example of a function that is its own inverse.
Find at least two other functions that are their own inverses.
Explain why they have this property.