Graphs of Inverse Functions

Understanding inverse functions and their graphs is a fundamental concept in Class 11 Maths. In maths, an inverse function is essentially a function that “reverses” another function. That is, if you start with a number, apply a function, and then apply its inverse function, you will get the original number back. In terms of graphs, it is about understanding how the function and its inverse are related visually. In this detailed exploration, we will delve into the concept of graphs of inverse functions.

Let us first understand what an inverse function is. Consider the function f(x). The inverse function, denoted as f-1(x), essentially performs the opposite operation of f(x). For the function and its inverse, the following condition will always be true:

f(f-1(x)) = x

and this too:

f-1(f(x)) = x

Not all functions have inverses. For a function to have an inverse, it must be one-to-one, meaning that each y-value is associated with one and only one x-value.

Example of an inverse function

Consider the function f(x) = 2x + 3. Let's find its inverse.

1. Start by substituting f(x) for y.

   y = 2x + 3

2. Solve for the value of x in terms of y.

   y - 3 = 2x
   x = (y - 3)/2

3. Replace y with f-1(x).

   f-1(x) = (x - 3)/2

So, the inverse of the function f(x) = 2x + 3 is f-1(x) = (x - 3)/2.

Graph of a function and its inverse

The graph of the inverse function can be viewed as a reflection of the graph of the original function over the line y = x. This line is known as the line of symmetry for the function and its inverse.

In the graph above, the red line represents the function f(x) = 2x + 3, the green line represents its inverse f-1(x) = (x - 3)/2, and the blue dashed line is the line y = x. You can see that the red and green lines are reflections of each other on the line y = x.

Steps to graph an inverse function

To graph an inverse function, follow these steps:

1. Recognize that the function is one-to-one.
2. Draw a graph of the original function.
3. Draw the line y = x for reference.
4. Reflect each point of the original function over the line y = x to find the corresponding points on the graph of the inverse.

Textual examples and verification

Let's take another example: If f(x) = x2, can we find its inverse? First, note that x2 is not a one-to-one function because both positive and negative values will give the same result when squared (for example, f(-2) = 4 and f(2) = 4), so it does not have an inverse over all real numbers.

However, if the domain is restricted to non-negative values (i.e., x ≥ 0), then the inverse function will be f-1(x) √x. We confirm that:

f(f-1(x)) = f(√x) = (√x)2 = x

f-1(f(x)) = f-1(x2) = √(x2) = x only if x ≥ 0

When defining inverses of non-one-to-one functions it is important to ensure matching of domain and range by restricting them appropriately.

Applications in the real world

Understanding inverse functions is also essential for real-world applications. Consider problems involving physics and engineering - the back-and-forth conversion between units can be modeled as an inverse function. For example, converting Celsius to Fahrenheit and vice versa can be accomplished using inverse functions and their inverses.

given:

F = (9/5)C + 32

To convert Fahrenheit back to Celsius, solve the above equation for C:

C = (5/9)(F - 32)

Here, the function F = (9/5)C + 32 and its inverse C = (5/9)(F - 32) ensure that we can vary the temperature seamlessly in either direction.

Conclusion

Graphs of inverse functions allow us to visualize how functions can be reversed, and how they are reflected on the line y = x. Through practice and understanding, this concept becomes helpful in solving mathematical equations and real-life problems.

Mastering inverse functions involves accepting their one-to-one nature, understanding their algebraic manipulation, and ensuring clarity in graphical representation. As you progress in mathematics, inverse functions will be repeated through more complex topics and scenarios.

Comments