Lesson Notes By Weeks and Term v5 - Grade 12

Lesson Video

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Performance objectives

• Find the inverse of linear functions.
• Find the inverse of quadratic functions (with domain restrictions).
• Find the inverse of exponential functions.
• Sketch graphs of functions and their inverses.
• Understand the relationship between the domain and range of a function and its inverse.

Lesson summary

Functions and their inverses are fundamental concepts in mathematics. Understanding them is crucial, not just for succeeding in further mathematical studies like Calculus and Linear Algebra, but also for various real-world applications. From modeling population growth to understanding financial trends and even designing efficient transportation routes, functions are everywhere. In South Africa, where we face challenges like resource management and economic inequality, a strong grasp of functions allows us to analyze data, predict outcomes, and develop informed solutions.

Lesson notes

What is a Function? A function is a relationship between two sets, called the domain and the range. For every element in the domain, there is exactly one corresponding element in the range.

Think of it as a machine: you put something in (the input – an element from the domain), and it spits something else out (the output – an element from the range). We often write a function as f(x), where x is the input and f(x) is the output. What is an Inverse Function? The inverse function, denoted by f -1 (x), "undoes" what the original function f(x) does. If f(a) = b, then f -1 (b) = a. In simpler terms, if you input a into the function f(x) and get b, then inputting b into the inverse function f -1 (x) will give you a. Important

Note: Not all functions have inverses. For a function to have an inverse, it must be a one-to-one function. This means that each element in the range corresponds to only one element in the domain (horizontal line test). Finding the Inverse Function Algebraically: The general steps to find the inverse of a function f(x) are: Replace f(x) with y. Swap x and y. Solve for y. Replace y with f -1 (x).

Graphical Representation: The graph of a function and its inverse are reflections of each other across the line y = x. This means that if the point (a, b) lies on the graph of f(x), then the point (b, a) lies on the graph of f -1 (x).

Inverses of Different Types of Functions: Linear Functions: Linear functions are of the form f(x) = mx + c, where m and c are constants and m ≠

0. These functions always have an inverse.

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

3. Replace f(x) with y: y = 2x + 3 Swap 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 Quadratic Functions: Quadratic functions are of the form f(x) = ax 2 + bx + c, where a ≠

0. These functions do not have an inverse over their entire domain because they are not one-to-one. To find an inverse, we must restrict the domain. We usually restrict the domain to x ≥ -b/2a or x ≤ -b/2a (where x = -b/2a is the axis of symmetry of the parabola).

Example: Find the inverse of f(x) = x 2 - 4 for x ≥

0. Replace f(x) with y: y = x 2 - 4 Swap x and y: x = y 2 - 4 Solve for y: x + 4 = y 2 => y = ±√(x + 4) Since we restricted the domain to x ≥ 0, the range of the inverse must be y ≥

0. Therefore, we choose the positive square root: y = √(x + 4)

Replace y with f -1 (x): f -1 (x) = √(x + 4) for x ≥ -4 Exponential Functions: Exponential functions are of the form f(x) = a x , where a > 0 and a ≠

1. The inverse of an exponential function is a logarithmic function.

Example: Find the inverse of f(x) = 2 x *.

Replace f(x) with y: y = 2 x Swap x and y: x = 2 y Solve for y: Using logarithms, y = log 2 (x)

Replace y with f -1 (x): f -1 (x) = log 2 (x) Domain and Range of Inverses 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).