Comprehensive Notes on the Inverse of Functions

Definition of the Inverse of a Function

The inverse of a function \( f(x) \), denoted as \( f^{-1}(x) \), is a function that reverses the operation of \( f(x) \). If \( f(a) = b \), then \( f^{-1}(b) = a \). In other words:

\[ f(f^{-1}(x)) = x \quad \text{and} \quad f^{-1}(f(x)) = x \]

For a function to have an inverse, it must be bijective—both one-to-one (injective) and onto (surjective).

Finding the Inverse of a Function

1. Replace \( f(x) \) with \( y \): \( y = f(x) \).
2. Swap \( x \) and \( y \): \( x = f^{-1}(y) \).
3. Solve for \( y \) in terms of \( x \).
4. Replace \( y \) with \( f^{-1}(x) \).

Examples of Finding Inverses

1. Inverse of Ordered Pairs

Consider the function defined by the set of ordered pairs: \[ f(x) = \{(2, 3), (5, 7), (-1, 10)\}. \] The inverse is obtained by swapping the \( x \)- and \( y \)-coordinates: \[ f^{-1}(x) = \{(3, 2), (7, 5), (10, -1)\}. \]

2. Inverse of an Exponential Function

Let \( f(x) = 3^x \). To find \( f^{-1}(x) \):

1. Write \( y = 3^x \).
2. Swap \( x \) and \( y \): \( x = 3^y \).
3. Solve for \( y \) using logarithms: \[ y = \log_3(x) \]
4. The inverse is: \[ f^{-1}(x) = \log_3(x), \quad x > 0 \]

3. Inverse of a Logarithmic Function

Let \( f(x) = \log_3(x) \). Its inverse is the exponential function: \[ f^{-1}(x) = 3^x, \quad x > 0 \]

4. Inverse of a Square Root Function

Let \( f(x) = \sqrt{3x - 9} \). To find \( f^{-1}(x) \):

1. Write \( y = \sqrt{3x - 9} \).
2. Square both sides: \[ y^2 = 3x - 9 \]
3. Solve for \( x \): \[ 3x = y^2 + 9 \implies x = \frac{y^2 + 9}{3} \]
4. Replace \( y \) with \( f^{-1}(x) \): \[ f^{-1}(x) = \frac{x^2 + 9}{3}, \quad x \geq 0 \]

5. Inverse of a Quadratic Function

Let \( f(x) = 2x^2 + 8x - 3 \). To find \( f^{-1}(x) \), we use the method of completing the square:

1. Write \( y = 2x^2 + 8x - 3 \).
2. Move the constant term: \[ y + 3 = 2x^2 + 8x \]
3. Factor out 2: \[ y + 3 = 2(x^2 + 4x) \]
4. Complete the square inside the parentheses: \[ x^2 + 4x \to (x + 2)^2 - 4 \] Substituting back: \[ y + 3 = 2((x + 2)^2 - 4) \] \[ y + 3 = 2(x + 2)^2 - 8 \]
5. Simplify: \[ y + 11 = 2(x + 2)^2 \]
6. Solve for \( x \): \[ (x + 2)^2 = \frac{y + 11}{2} \implies x + 2 = \pm \sqrt{\frac{y + 11}{2}} \] \[ x = -2 \pm \sqrt{\frac{y + 11}{2}} \]
7. Select the branch of the inverse based on the domain. For example, if \( x \geq -2 \): \[ f^{-1}(x) = -2 + \sqrt{\frac{x + 11}{2}} \]

6. Practical Example: Temperature Conversion

The formula for converting Celsius to Fahrenheit is: \[ F = \frac{9}{5}C + 32 \] To find the inverse (Fahrenheit to Celsius):

1. Write \( F = \frac{9}{5}C + 32 \).
2. Swap \( F \) and \( C \): \( C = \frac{9}{5}F + 32 \).
3. Solve for \( C \): \[ F - 32 = \frac{9}{5}C \implies C = \frac{5}{9}(F - 32) \]
4. The inverse is: \[ C = \frac{5}{9}(F - 32) \]

Practice Questions

1. Find the inverse of \( f(x) = 4x + 5 \).
2. Find the inverse of \( f(x) = \sqrt{x + 7} \).
3. Find the inverse of \( f(x) = \frac{2x + 3}{x - 4} \).
4. Find the inverse of \( f(x) = \ln(x - 2) \).
5. Find the inverse of \( f(x) = 2x^2 + 12x + 7 \) by completing the square.

Answers to Practice Questions

1. \( f^{-1}(x) = \frac{x - 5}{4} \)
2. \( f^{-1}(x) = x^2 - 7, \quad x \geq 0 \)
3. \( f^{-1}(x) = \frac{3x + 4}{x - 2} \)
4. \( f^{-1}(x) = e^x + 2 \)
5. \( f^{-1}(x) = -3 + \sqrt{\frac{x - 5}{2}} \)