Understanding Inverse Functions: A Step-by-Step Guide and Example

Inverse Function

An inverse function, denoted as f^(-1), is the reverse operation of a given function f

An inverse function, denoted as f^(-1), is the reverse operation of a given function f. It reverts the output values of f back to the original input values. In other words, if a function f takes an input x and produces an output y, the inverse function f^(-1) takes y as input and produces x as output.

To find the inverse function of a given function, you typically follow these steps:

1. Start with the original function f(x).
2. Replace f(x) with y.
3. Swap the x and y variables in the equation, so the equation becomes x = f^(-1)(y).
4. Solve the equation for y in terms of x. This will give you the equation for the inverse function f^(-1)(x).

Let’s work through an example to illustrate this process:

Example: Find the inverse function of f(x) = 2x + 3.

Step 1: Start with the original function f(x) = 2x + 3.

Step 2: Replace f(x) with y, so we have y = 2x + 3.

Step 3: Swap the x and y variables, giving x = 2y + 3.

Step 4: Solve the equation for y in terms of x.
To do this, first subtract 3 from both sides:
x – 3 = 2y.

Next, divide both sides by 2:
(x – 3) / 2 = y.

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

To verify that a function and its inverse are indeed inverses, you can compose them together. If f(x) = 2x + 3 and its inverse is f^(-1)(x) = (x – 3) / 2, then f(f^(-1)(x)) should equal x, and f^(-1)(f(x)) should also equal x for any x.

Let’s check this with our example:

f(f^(-1)(x)) = f((x – 3) / 2)
= 2 * ((x – 3) / 2) + 3
= (x – 3) + 3
= x.

f^(-1)(f(x)) = f^(-1)(2x + 3)
= (2x + 3 – 3) / 2
= 2x / 2
= x.

Since both compositions equal x, it confirms that f(x) = 2x + 3 and f^(-1)(x) = (x – 3) / 2 are indeed inverses of each other.

I hope this explanation helps in understanding inverse functions. Let me know if you have any further questions!

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