Finding and Evaluating Inverse Functions

1. What is an Inverse Function?

An inverse function does the exact opposite of the original function. If a function f turns an input of 3 into an output of 10, then its inverse, written as f⁻¹(x), will turn an input of 10 back into the original 3.

It's like reversing a function machine. If the original machine multiplies by 7 and then adds 2, the inverse machine will first subtract 2 and then divide by 7—the inverse operations in the reverse order.

Important Note on Notation: The notation f⁻¹(x) means "the inverse of function f". The -1 is not an exponent. It does not mean $1/f(x)$. This is a very common point of confusion, so be mindful of it.

To see this concept visualised clearly with the function machine analogy, watch the beginning of the following video.

2. The Method for Finding an Inverse Function

While the function machine is great for understanding the concept, you need a reliable algebraic method for your exam. There is a standard four-step process that works for all the inverse function questions you will encounter.

Here are the steps in detail:

1. Replace f(x) with y: Rewrite the function as an equation, e.g., if $f(x)=2x+1$, write $y=2x+1$.
2. Swap x and y: Replace every x with a y and every y with an x. This is the key step that represents the "inverting" of inputs and outputs. The equation becomes $x=2y+1$.
3. Make y the subject: Rearrange the new equation to isolate y. This is where your algebraic skills are tested. $$\begin{array}{rl}x& =2y+1\\ x-1& =2y\\ \frac{x-1}{2}& =y\end{array}$$
4. Replace y with f⁻¹(x): Rewrite the final expression in proper inverse function notation: $f^{-1}(x)=\frac{x-1}{2}$.

This method is reliable and will earn you method marks in the exam, even if you make a small slip in the final rearrangement.

To see this method applied in a clear, narrated format, the same video provides a great walkthrough.

3. Handling More Complex Rearrangements

The difficulty in inverse function questions often comes from Step 3: rearranging the equation. You've mentioned that choosing the most efficient method and avoiding errors in multi-step problems is an area you want to improve. Let's look at two common, more challenging scenarios.

Case 1: The subject appears twice (Rational Functions)

Consider a function like $f(x)=\frac{2x+5}{3x-1}$. When you swap x and y, you get $x=\frac{2y+5}{3y-1}$. To make y the subject, you need to collect all y terms and factorize. This is a crucial technique.

The following video from The Organic Chemistry Tutor demonstrates this process very effectively. While the video covers some advanced concepts, the segment on this specific type of problem is perfectly relevant and clear.

Case 2: Functions with Powers and Roots

If your function involves squares or square roots, your rearrangement will involve the inverse operations: square rooting or squaring.

For a function like $f(x)=x^2-4$ where $x\ge 0$, the process would be:

1. $y=x^2-4$
2. $x=y^2-4$
3. $x+4=y^2$
4. $\sqrt{x+4}=y$
5. $f^{-1}(x)=\sqrt{x+4}$

Note: An unrestricted parabola $y=x^2-4$ is not a "one-to-one" function (it fails the horizontal line test), so it doesn't have a true inverse function. That's why exam questions will specify a domain like $x\ge 0$ to make it one-to-one. For IGCSE, you typically only need to consider the positive root unless told otherwise.

The 1st Class Maths video also has a good, concise example of this.

4. Evaluating an Inverse Function

Often, a question will ask you to find a value like f⁻¹(5). There are two ways to approach this, and choosing the right one can save you time.

Method 1: Find the inverse function first (The Standard Method)

1. Find the expression for $f^{-1}(x)$ using the four-step method.
2. Substitute the given value into your inverse function. For example, if $f^{-1}(x)=\frac{x-1}{2}$, then $f^{-1}(5)=\frac{5-1}{2}=2$.

This method is robust and always works. The video below demonstrates this approach with a more complex example.

Method 2: The Efficiency Shortcut
Remember that if f(x) = y, then f⁻¹(y) = x.
To find f⁻¹(5), we are looking for the input x that makes the original function output 5.
So, you can solve the equation f(x) = 5.

Let's use our earlier example, $f(x)=2x+1$. To find f⁻¹(5):
Set $f(x)=5$:

So, f⁻¹(5) = 2. This gives the same answer as Method 1, but can be much faster if the question only asks for the value and not the expression for f⁻¹(x). Knowing both methods gives you a strategic advantage.

Conclusion

Well done. You have now covered the entire process for finding and using inverse functions, from the basic concept to the algebraic methods required for exam success.

Key Takeaways:

• An inverse function f⁻¹(x) reverses the action of the original function f(x).
• The notation f⁻¹(x) does not mean $1/f(x)$.
• The most reliable method is the four-step process: (1) Write y=f(x), (2) Swap x and y, (3) Rearrange for y, (4) Write as f⁻¹(x).
• For evaluating f⁻¹(a), you can either find f⁻¹(x) first and substitute x=a, or solve the equation f(x) = a.

Preview of the Next Lesson:
We have now concluded our main work on functions. In the next lesson, we will shift our focus to the other part of this module's title: sequences. We will begin by looking at a specific type of sequence known as a geometric sequence, learning how to recognize them and find their nth term.