Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Define an inverse function and its notation (f⁻¹(x)).
• Use the horizontal line test to see if a function has an inverse.
• Calculate the inverse of a function algebraically.
• Sketch the graphs of functions and their inverses.
• Determine the domain and range of an inverse function.

Lesson summary

This week, we delve deeper into the fascinating world of functions, focusing on inverse functions and their graphs. Understanding inverse functions is crucial not only for your mathematical journey but also for various fields like engineering, economics, and computer science. For example, understanding how a cellphone company maps data usage to cost (a function) allows them to accurately bill you. Inverse functions would allow you to calculate how much data you used based on a specific bill amount. This topic builds on your existing knowledge of functions, focusing on reversing the process and understanding the conditions under which this is possible.

Lesson notes

What is an Inverse Function? An inverse function is essentially the "reverse" of a given function. If a function f takes an input x and produces an output y, then the inverse function, denoted as f⁻¹, takes y as input and produces x as output. In other words, if f(x) = y, then f⁻¹(y) = x.

Important Notes: The "-1" in f⁻¹(x)* is NOT an exponent. It represents the inverse function. Not all functions have inverses. For a function to have an inverse, it must be one-to-one. One-to-One Functions A function is one-to-one (or injective) if each element of the range (output) corresponds to exactly one element of the domain (input).

Graphical Test: The Horizontal Line Test A function is one-to-one if and only if no horizontal line intersects its graph more than once. If a horizontal line intersects the graph more than once, it means that two different x-values map to the same y-value, and thus the function is not one-to-one.

Algebraic Test: To prove a function f(x) is one-to-one algebraically, assume f(a) = f(b) for some a and b in the domain of f. If you can show that a = b, then the function is one-to-one.

Example: Show that f(x) = 2x + 3 is one-to-one. Assume f(a) = f(b).

Then: 2a + 3 = 2b + 3 2a = 2b a = b Therefore, f(x) = 2x + 3 is one-to-one. Finding the Inverse Function Algebraically Here's the step-by-step process to find the inverse of a function f(x): Replace f(x) with y. This makes the algebraic manipulation easier. Swap x and y. This is the crucial step in reversing the function. Solve for y in terms of x. Isolate y on one side of the equation. Replace y with f⁻¹(x). This is the notation for the inverse function. State any restrictions on the domain of f⁻¹(x), based on its formula, and relating to the range of f(x).