Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Determine the inverse of a given function.
• Sketch the graphs of inverse functions.
• Analyze the relationship between the domain and range of a function and its inverse.
• Apply inverse functions to solve problems.

Lesson summary

This week, we delve deeper into the fascinating world of functions, building upon the foundational knowledge acquired last week. We move beyond identifying functions to analyzing and manipulating them. Understanding functions is crucial, not just for further mathematical studies but also for interpreting real-world data and making informed decisions. From predicting electricity consumption based on temperature to modelling population growth, functions are the backbone of many practical applications. For South African learners, a strong understanding of functions opens doors to careers in STEM fields vital for the country's development.

Lesson notes

This week focuses on inverse functions.

Definition of an Inverse Function: An inverse function "undoes" what the original function does. If a function f takes an input x and produces an output y (i.e., f(x) = y), then the inverse function, denoted as f -1 , takes y as its input and returns x as its output (i.e., f -1 (y) = x).

Key Properties and Requirements: One-to-One Functions: For a function to have an inverse, it must be a one-to-one function. A function is one-to-one if each x-value maps to a unique y-value, and each y-value is associated with a unique x-value. Graphically, this means the function passes the horizontal line test (no horizontal line intersects the graph more than once). If a function is not one-to-one, you may be able to restrict its domain to make it one-to-one and then find its inverse.

Finding the Inverse: To find the inverse of a function f(x): Replace f(x) with y. Swap x and y. Solve the new equation for y. Replace y with f -1 (x)*.

Domain and Range: The domain of f(x) becomes the range of f -1 (x), and the range of f(x) becomes the domain of f -1 (x). This is a fundamental relationship.

Graphical Representation: The graph of f -1 (x) is a reflection of the graph of f(x) across the line y = x. This provides a visual way to understand inverse functions.

Worked example

Example 1: Find the inverse of f(x) = 2x +

3. Step 1: Replace f(x) with y: y = 2x + 3

Step 2: Swap x and y: x = 2y + 3

Step 3: Solve for y:

x - 3 = 2y

y = (x - 3) / 2

Step 4: Replace y with f -1 (x): f -1 (x) = (x - 3) / 2

Therefore, the inverse of f(x) = 2x + 3 is f -1 (x) = (x - 3) /

2. Example 2: Determine the inverse of g(x) = x 2 , where x ≥ 0. (Note the restricted domain).