Lesson Notes By Weeks and Term v5 - Grade 12

Lesson Video

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Performance objectives

• Use the horizontal line test to see if a function has an inverse.
• Find the inverse of linear, quadratic, and exponential functions.
• Sketch the graphs of a function and its inverse.
• Determine the domain and range of an inverse function.

Lesson summary

Functions are the building blocks of mathematical modeling, allowing us to describe relationships between quantities. This week, we delve into the concept of inverse functions, which essentially "undo" the action of a function. Understanding inverses is crucial because it enables us to solve equations, analyze mathematical models from different perspectives, and appreciate the symmetrical nature of certain relationships. Think about currency conversion. If you're travelling overseas and want to convert Rands to Euros, that's a function. Going the other way, converting Euros back to Rands, that's the inverse function. Similarly, consider a cellphone data package.

Lesson notes

What is an Inverse Function? An inverse function, denoted as f⁻¹(x), is a function that "reverses" the effect of the original function, f(x). In other words, if f(a) = b, then f⁻¹(b) = a. Not all functions have inverses. For a function to have an inverse, it must be a one-to-one function. One-to-One Functions and the Horizontal Line Test: A function is one-to-one if each y-value corresponds to only one x-value. Graphically, this means that no horizontal line intersects the graph of the function more than once. This is called the Horizontal Line Test. If a function passes the horizontal line test, it is one-to-one and has an inverse. If it fails, it does not. Finding the Inverse Function Algebraically: Replace f(x) with y: This simplifies the notation.

Interchange x and y: This is the key step in finding the inverse.

Solve for y: Isolate y to express it in terms of x. Replace y with f⁻¹(x): This is the notation for the inverse function. Important Considerations for Quadratic Functions: Quadratic functions, like f(x) = x², are not one-to-one over their entire domain (because both x and -x will yield the same f(x)).

Therefore, to find an inverse, we need to restrict the domain. We typically restrict the domain to x ≥ 0 or x ≤ 0 to make the function one-to-one.

Graphs of Functions and Their Inverses: The graph of f⁻¹(x) is a reflection of the graph of f(x) about the line y = x. This is because the x and y coordinates are swapped in the inverse function.

Domain and Range of Inverse Functions: The domain of f⁻¹(x) is the range of f(x), and the range of f⁻¹(x) is the domain of f(x). This is a direct consequence of the inverse function "undoing" the original function.

Worked example

Example 1: Linear Function

Let f(x) = 2x +

3. Find f⁻¹(x).

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

Interchange x and y: x = 2y + 3

Solve for y:

x - 3 = 2y

y = (x - 3)/2

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

Why does this work? Let's test. f(2) = 2(2) + 3 =

7. Now, f⁻¹(7) = (7-3)/2 =

2. The inverse correctly "undoes" the original function.

Example 2: Quadratic Function (with restricted domain)