Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Determine the inverse of a linear or quadratic function.
• Sketch graphs of exponential functions like y = a^x + q.
• Determine the equation of an exponential function from its graph.
• Understand the relationship between a function and its inverse.

Lesson summary

This week, we delve deeper into the fascinating world of functions. Building on your Grade 10 knowledge, we will specifically focus on inverse functions and exponential functions. Understanding these types of functions is crucial not only for further mathematical studies but also for modeling real-world phenomena. For example, exponential functions are used to model population growth, compound interest, and the decay of radioactive materials (relevant in understanding the environmental impact of mining, for instance). Inverse functions are essential in cryptography and in understanding how to reverse processes – a concept useful in many fields.

Lesson notes

2.1 Inverse Functions An inverse function "undoes" what the original function does. If a function f takes an input x and produces an output y, then the inverse function f -1 takes y as input and produces x as output. Symbolically, if f(x) = y, then f -1 (y) = x.

Finding the Inverse: Replace f(x) with y: This simplifies the notation.

Swap x and y: This represents the "undoing" process.

Solve for y: Isolate y in terms of x.

Replace y with f -1 (x): This indicates that you have found the inverse function.

Important Considerations: Not all functions have inverses: For a function to have an inverse, it must be one-to-one. This means that each x-value maps to a unique y-value, and each y-value maps to a unique x-value (passes the horizontal line test). Quadratic functions, for example, are not one-to-one over their entire domain.

Therefore, we often restrict the domain of quadratic functions to ensure they have inverses.

Domain and Range: The domain of f becomes the range of f -1 , and the range of f becomes the domain of f -1 *.

Graphical Representation: The graph of f -1 (x) is a reflection of the graph of f(x) across the line y = x.

Worked example

Example 1: Linear Function

Find the inverse of f(x) = 2x + 3.

y = 2x + 3

x = 2y + 3

x - 3 = 2y

y = (x - 3) / 2

f -1 (x) = (x - 3) / 2

Explanation: We swapped x and y and then solved for y. The inverse function f -1 (x) takes an input, subtracts 3, and then divides by

2. Example 2: Quadratic Function (with Domain Restriction)