Lesson Notes By Weeks and Term v5 - Grade 12

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Performance objectives

• Find the inverse of a function algebraically.
• Sketch the graph of an inverse function.
• Understand the relationship between the domain and range of a function and its inverse.
• Verify if two functions are inverses of each other.

Lesson summary

Functions are the fundamental building blocks of mathematical modelling. Understanding functions and their inverses is crucial for solving real-world problems, from calculating profit margins in a small business to predicting population growth in a community. This week, we will delve deeper into the concept of inverse functions, focusing on how to determine them algebraically and graphically, and exploring their properties. Understanding inverses allows us to "undo" a function, which is extremely useful in various applications. For example, if a function calculates the cost of producing x items, its inverse would tell you how many items can be produced for a given cost.

Lesson notes

What is an Inverse Function? An inverse function, denoted as f⁻¹(x), "undoes" the action of the original function f(x). In other words, if f(a) = b, then f⁻¹(b) = a. Not all functions have an inverse. A function must be one-to-one (each x-value corresponds to a unique y-value, and each y-value corresponds to a unique x-value) to have an inverse. Graphically, a one-to-one function passes both the vertical and horizontal line tests. Finding the Inverse Algebraically Replace f(x) with y: This simplifies the equation.

Swap x and y: This reflects the inverse relationship.

Solve for y: Isolate y to express the inverse function in terms of x. Replace y with f⁻¹(x): This denotes the inverse function.

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 Verification: To verify, we can compose the function and its inverse. f(f⁻¹(x)) should equal x, and f⁻¹(f(x)) should also equal x. f(f⁻¹(x)) = 2((x - 3) / 2) + 3 = (x - 3) + 3 = x f⁻¹(f(x)) = ((2x + 3) - 3) / 2 = (2x) / 2 = x Since both compositions result in x, we have verified that f⁻¹(x) = (x - 3) / 2 is the inverse of f(x) = 2x +

3. Example 2: Quadratic Function (with restricted domain) Find the inverse of g(x) = x² - 4, where x ≥ 0. y = x² - 4 x = y² - 4 x + 4 = y² y = ±√(x + 4) Since the original function was defined for x ≥ 0, the range of the inverse must be y ≥

0. Therefore, we take the positive square root: g⁻¹(x) = √(x + 4)

Note: The domain restriction x ≥ 0 is crucial. Without it, the original quadratic function is not one-to-one, and we cannot find a unique inverse. The range of the original function becomes the domain of the inverse, and vice versa. The range of g(x) is y ≥ -4, so the domain of g⁻¹(x) is x ≥ -

4. Example 3: Exponential Function Find the inverse of h(x) = 3ˣ. y = 3ˣ x = 3ʸ To solve for y, we need to use logarithms: y = log₃(x) h⁻¹(x) = log₃(x)

Note: The base of the exponential function becomes the base of the logarithmic function. The domain of h(x) is all real numbers, and its range is y >

0. Therefore, the domain of h⁻¹(x) is x > 0, and its range is all real numbers. Graphical Representation of Inverse Functions The graph of an inverse function is a reflection of the original function across the line y = x. If you have the graph of f(x), you can sketch the graph of f⁻¹(x) by reflecting it across this line. Domain and Range of Inverse Functions The domain of f(x) is the range of f⁻¹(x), and the range of f(x) is the domain of f⁻¹(x). This is a fundamental property of inverse functions. Composition of Functions and Inverses As shown in the previous examples, the composition of a function and its inverse results in the identity function, x. This is a key property for verifying that two functions are indeed inverses of each other.

Formally: f(f⁻¹(x)) = x and f⁻¹(f(x)) = x for all x in the appropriate domains. Guided Practice (With Solutions)

Question 1: Find the inverse of f(x) = -x + 5 and verify your answer.

Solution: y = -x + 5 x = -y + 5 x - 5 = -y y = -x + 5 f⁻¹(x) = -x + 5 Verification: f(f⁻¹(x)) = -(-x + 5) + 5 = x - 5 + 5 = x f⁻¹(f(x)) = -( -x + 5) + 5 = x - 5 + 5 = x Therefore, f⁻¹(x) = -x +

5. In this specific case, the function is its own inverse.

Question 2: Find the inverse of g(x) = (x - 2)², x ≥

2. Solution: y = (x - 2)² x = (y - 2)² ±√x = y - 2 y = 2 ± √x Since x ≥ 2, the range of the inverse must be y ≥

2. Therefore: g⁻¹(x) = 2 + √x

Commentary: We choose the positive square root to satisfy the domain restriction of the original function. The domain of the inverse is x ≥

0. Question 3: Find the inverse of h(x) = 5ˣ -

1. Solution: y = 5ˣ - 1 x = 5ʸ - 1 x + 1 = 5ʸ y = log₅(x + 1) h⁻¹(x) = log₅(x + 1)

Commentary: Remember to add 1 to x before taking the logarithm. The domain of h⁻¹(x) is x > -

1. Question 4: Determine if f(x) = 3x - 1 and g(x) = (x + 1)/3 are inverses of each other.

Solution: We need to check f(g(x)) and g(f(x)). f(g(x)) = 3((x + 1)/3) - 1 = (x + 1) - 1 = x g(f(x)) = ((3x - 1) + 1) / 3 = (3x) / 3 = x Since both compositions equal x, f(x) and g(x) are inverses of each other.

Question 5: If f(x) = √(x - 3), x ≥ 3, find f⁻¹(x) and state its domain and range.

Solution: y = √(x - 3) x = √(y - 3) x² = y - 3 y = x² + 3 f⁻¹(x) = x² + 3 The domain of f⁻¹(x) is x ≥ 0 (which is the range of f(x)), and the range is y ≥ 3 (which is the domain of f(x)). Independent Practice (Questions Only) Find the inverse of f(x) = 4x -

7. Find the inverse of g(x) = x² + 2, x ≤

0. Find the inverse of h(x) = 2ˣ +

3. Determine if f(x) = (x - 5)/2 and g(x) = 2x + 5 are inverses of each other. Given f(x) = √ (2x + 1), find f⁻¹(x) and state its domain and range. Find the inverse of f(x) = -3x -

2. Graph both the function and its inverse on the same set of axes. Find the inverse of g(x) = 1/x. What do you notice? The function p(x) = 1000(1.05)ˣ models the population of a small town in South Africa x years from now.