**Algebraic Method**

Finding the inverse of a function algebraically is a systematic process. Here's a step-by-step breakdown:

1. **Function Notation Replacement**:

- Start by replacing the function notation, f(x), with y. This makes the equation easier to manipulate.

2. **Interchanging x and y**:

- Swap the roles of x and y. This is the essence of finding the inverse; you're essentially switching the input and output values.

3.** Solving for y**:

- With the equation now having x and y swapped, rearrange the equation to isolate y on one side. This will give you the inverse function. Understanding the basics of domain and range is crucial in this step.

4. **Inverse Notation**:

- Once you've isolated y, replace it with the inverse notation f
^{(-1)}(x). This denotes the inverse function of f(x).

**Example**: To find the inverse of f(x) = 5x - 7:

- Start with y = 5x - 7.
- Swap x and y to get x = 5y - 7.
- Rearrange to solve for y to get y = (x + 7)/5.
- The inverse function is then f
^{(-1)}(x) = (x + 7)/5.

**Graphical Method**

Visualising functions and their inverses can be incredibly insightful. Here's how you can determine the inverse of a function using its graph:

1.** Sketch the Function**:

- Begin by plotting the graph of the function you're given. This is especially helpful when dealing with quadratic functions.

2.** Draw the Identity Line**:

- Plot the line y = x. This line will serve as a mirror. The graph of the inverse function will be a reflection across this line.

3.** Reflect the Function**:

- Take the graph of the function and reflect it over the line y = x. The resulting graph will represent the inverse function.

**Example**: For the function f(x) = x^{2} (which is a parabola opening upwards), its inverse will be the square root function. This is because the square root function is the reflection of the parabola over the line y = x.

**Properties of Inverse Functions**

Understanding the properties of inverse functions can provide deeper insights into their behaviour:

**Definition**: If you have a function and its inverse, applying the function followed by its inverse (or vice versa) will always return the original input. Mathematically, this is represented as f(f^{(-1)}(x)) = f^{(-1)}(f(x)) = x.**Reflection Property**: The graphs of a function and its inverse are symmetrical about the line y = x. This means that if a point (a, b) lies on the graph of the function, then the point (b, a) will lie on the graph of its inverse.**Uniqueness**: For a function to have an inverse, it must be bijective, meaning it's both injective (one-to-one) and surjective (onto). If a function doesn't meet these criteria, it won't have an inverse. This is often discussed in relation to rational functions.**Composition**: The composition of a function and its inverse always results in the identity function. This means that for every x in the domain, f(f^{(-1)}(x)) = x and f^{(-1)}(f(x)) = x.

**Real-World Application**

Inverse functions play a crucial role in various real-world scenarios. For instance:

**Cryptography**: The art of encoding and decoding messages can be thought of as using functions and their inverses. An encoding function might transform a message into a coded format, and its inverse would decode it back into the original message.**Engineering**: Many systems are designed with features that can "undo" previous actions. These can often be modelled using functions and their inverses. A solid understanding of exponential equations is beneficial in these contexts.

**Example Questions**

1. **Question**: Determine the inverse of the function f(x) = 4x + 9.

**Solution**:- Start with y = 4x + 9.
- Swap x and y to get x = 4y + 9.
- Solve for y to get y = (x - 9)/4.
- The inverse function is f
^{(-1)}(x) = (x - 9)/4.

2.** Question**: Given the function f(x) = x^{3}, sketch its graph and the graph of its inverse.

**Solution**: The graph of f(x) = x^{3}is a cubic curve. Its inverse is the cube root function, which can be visualised as the reflection of the cubic curve over the line y = x. Familiarity with radians can help in understanding the graphical representations of these functions.

By delving into the methods and properties of inverse functions, students can gain a comprehensive understanding of how functions can be "undone" and the significance of this concept in various mathematical and real-world contexts.

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.