CTJan27 Online Year 10 - Inverse Functions

Introduction to "Undoing"

Have you ever thought about how you put your shoes on? You put your socks on first, and then your shoes. If you want to reverse or "undo" this process, do you take your socks off first? No! You have to take your shoes off, and then your socks. Mathematics has a very similar idea for "undoing" actions, and we call this an "inverse". But before we can talk about inverse functions, we need to understand inverse relations.

What exactly is a relation? In simple terms, it is just a connection between inputs (x-values) and outputs (y-values). It is often written as a set of ordered pairs.

Comparing Inverses and Opposites

Think of a regular relation as a travel path from a starting town (the domain, or x-values) to an ending town (the range, or y-values). For example, the ordered pair $(2, 5)$ means "start at 2, travel and end up at 5".

So, what is the inverse relation? It is simply the exact reverse path! The inverse relation goes from the ending town back to the starting town. Therefore, the inverse of traveling $(2, 5)$ is traveling $(5, 2)$.

It is easy to confuse "inverse" with "opposite". Compare this reversing action to making a number negative. The opposite of 5 is -5. But an inverse relation has absolutely nothing to do with making a number negative. We do not change signs unless the original number already had a negative sign; we simply swap the starting and ending places!

The Golden Rules of Inverse Relations

To find an inverse relation, we follow these simple rules:

1. Swap the $x$ and $y$ coordinates of every ordered pair in the original relation.

2. The domain (all possible x-values) of the original relation becomes the range (all possible y-values) of the inverse relation.

3. The range of the original relation becomes the domain of the inverse relation.

Worked Example

Let's put this into practice.

Question: Find the inverse of the relation $R = \{(1, 4), (2, 5), (3, 6)\}$.

Step 1: Take the first pair $(1, 4)$ and swap the numbers' places to get $(4, 1)$.

Step 2: Swap the second pair $(2, 5)$ to get $(5, 2)$.

Step 3: Swap the third pair $(3, 6)$ to get $(6, 3)$.

Answer: The inverse relation is $\{(4, 1), (5, 2), (6, 3)\}$.

Your Turn

Question: Find the inverse of the relation $S = \{(-2, 7), (0, 8), (4, -1)\}$.

Hint: Just swap the $x$ and $y$ values! Do not change the positive or negative signs.

Answer: The inverse relation is $\{(7, -2), (8, 0), (-1, 4)\}$.

What If We Have an Equation?

What if the relation is given as an equation, like $y = 3x + 2$, instead of a list of points?

To find the equation of the inverse relation, you follow the exact same logic: physically swap the $x$ and $y$ variables in the equation.

The inverse relation is simply $x = 3y + 2$. Later on, you will learn how to rearrange this new equation to solve for $y$, but the fundamental first step is always the swap!

Can Every Function Be Undone?

Think about simple actions in your day. If you put your shoes on, you can take them off. You just undid the action! In math, we often want to "undo" a function. If a function gives us an output, can we trace it back to the exact input it came from?

To answer this, we need a special type of function.

One-to-One Functions

A regular function guarantees that every input ($x$) has exactly one output ($y$). But multiple inputs can share the same output. Think about $f(x) = x^2$. Both $x = 2$ and $x = -2$ give us $y = 4$.

If I tell you the output is $4$, can you tell me exactly what the input was? No, because it could be $2$ or $-2$. Because of this confusion, $f(x) = x^2$ cannot be perfectly undone.

A one-to-one function goes a step further. Not only does every $x$ have exactly one $y$, but every $y$ comes from exactly one $x$!

• Compare and Contrast: In $f(x) = x^2$, the output $4$ has two inputs ($2$ and $-2$). This is NOT one-to-one. But in $f(x) = x^3$, every output has exactly one unique input. The output $8$ only comes from $x = 2$. This IS one-to-one.

The Horizontal Line Test

We use the Vertical Line Test to see if a graph represents a function. But what if we want to visually check if a function is one-to-one?

Since a one-to-one function cannot repeat $y$-values, we use the Horizontal Line Test.

1. Imagine drawing perfectly flat horizontal lines across the graph of a function.
2. If any horizontal line touches the graph in more than one place, the function repeats a $y$-value. It is NOT one-to-one.
3. If every horizontal line touches the graph at most once, it IS one-to-one!

Inverse Function Notation

If a function is one-to-one, we can undo it. We call this "undoing" function the inverse function.

If our original function is named $f$, we write its inverse as $f^{-1}$. We read this as "$f$ inverse".

• Important Contrast: The $-1$ looks like an exponent, but it is NOT! $f^{-1}(x)$ means the inverse function. It does NOT mean $\frac{1}{f(x)}$ (which is the reciprocal).

If the original function $f$ takes an input of $a$ and gives an output of $b$, then $f(a) = b$.

The inverse function simply reverses this: it takes $b$ as the input and gives $a$ as the output. So, $f^{-1}(b) = a$.

Worked Example & Your Turn

Example: Let $f$ be a one-to-one function where $f(5) = 12$. What is $f^{-1}(12)$?

Step 1: Understand the original relationship. The function $f$ turns $5$ into $12$.

Step 2: Apply the inverse concept. The inverse function $f^{-1}$ takes the output ($12$) and brings it back to the input ($5$).

Answer: $f^{-1}(12) = 5$.

Your Turn: Let $g$ be a one-to-one function where $g(-3) = 7$. What is $g^{-1}(7)$? (Answer: -3)

What is an Inverse Function?

Have you ever tied your shoes, only to immediately untie them? What happens when you do that? You end up exactly where you started! In math, an inverse function does exactly the same thing. It is a rule that reverses, or "undoes", the effect of another function.

If a function $f$ takes an input $x$ and turns it into an output $y$, the inverse function (written as $f^{-1}$) takes that $y$ and turns it back into $x$.

Finding Inverses from Ordered Pairs and Tables

How do we represent reversing a process with points? Think about an ordered pair $(x, y)$. If the original function turns $x$ into $y$, the inverse must turn $y$ into $x$.

Therefore, to find the inverse of a set of ordered pairs, we simply swap the $x$ and $y$ values.

Compare & Contrast:

• Original Function: The set of points $\{(1, 5), (2, 10), (3, 15)\}$. The domain (inputs) is $\{1, 2, 3\}$.
• Inverse Function: The set of points $\{(5, 1), (10, 2), (15, 3)\}$. The domain (inputs) is now $\{5, 10, 15\}$.

Notice how the domain of the original function becomes the range of the inverse, and vice-versa!

Worked Example:

Given the table for $f(x)$ below, find $f^{-1}(8)$.

Your Turn: Look at the table. $f(x)$ outputs $8$ when the input is $3$. So, the inverse function takes the input $8$ and outputs $3$. Therefore, $f^{-1}(8) = 3$.

Steps to Algebraically Find an Inverse Function

What if you are given an equation instead of a table? How do you mathematically "swap" the inputs and outputs to find the inverse formula? We can break it down into four simple steps:

1. Replace $f(x)$ with $y$: This just makes the equation easier to write and work with.
2. Swap $x$ and $y$: This is the core idea of an inverse! We are literally trading the input and output.
3. Solve for the new $y$: Use algebra to get $y$ by itself again.
4. Replace $y$ with $f^{-1}(x)$: This is the proper notation to show that you have found the inverse function.

Let's see this in action. Suppose $f(x) = 2x + 6$.

1. Replace: $y = 2x + 6$
2. Swap: $x = 2y + 6$
3. Solve: Subtract $6$ to get $x - 6 = 2y$. Then divide by $2$ to get $\frac{x - 6}{2} = y$.
4. Replace: $f^{-1}(x) = \frac{x - 6}{2}$

How Can We Be Sure Two Functions Are Inverses?

Imagine you lock a door, and then you use a key to unlock it. You are back exactly where you started: with an unlocked door.

If a friend gives you a key, how can you be absolutely sure it is the correct inverse key? You would test it! You would lock the door, use the key, and see if the door unlocks.

In math, we use a very similar test. If we have a function $f(x)$ and we think another function $g(x)$ is its inverse, we test them by putting them together. We call this process composition.

The Composition Test

When we compose functions, we put one function inside the other.

If $f(x)$ and $g(x)$ are truly inverses, applying $f$ and then applying $g$ should leave our original input $x$ completely unchanged.

The rule is: Two functions $f(x)$ and $g(x)$ are inverses if and only if:

1. $f(g(x)) = x$
2. $g(f(x)) = x$

You must check both directions!

Compare and Contrast:

• Checking a single number: $f(g(2)) = 2$ tells you they might be inverses for that one number.
• Checking the function: $f(g(x)) = x$ tells you they are inverses for every possible number. We always use $x$ to prove it for all numbers.

Worked Example: Verifying Inverses

Question: Verify whether $f(x) = 3x - 12$ and $g(x) = \frac{1}{3}x + 4$ are inverse functions.

Step 1: Find $f(g(x))$. Substitute the entire function $g(x)$ into the $x$ spot in $f(x)$. $f(g(x)) = 3(\frac{1}{3}x + 4) - 12$

Step 2: Simplify the expression. Distribute the $3$. $f(g(x)) = (1x + 12) - 12$ $f(g(x)) = x$

Step 3: Now check the other direction, $g(f(x))$. Substitute $f(x)$ into $g(x)$. $g(f(x)) = \frac{1}{3}(3x - 12) + 4$

Step 4: Simplify. Distribute the $\frac{1}{3}$. $g(f(x)) = (x - 4) + 4$ $g(f(x)) = x$

Answer: Since both $f(g(x)) = x$ and $g(f(x)) = x$, they are inverse functions!

Your Turn

Question: Verify if $f(x) = 2x + 8$ and $g(x) = \frac{x}{2} - 4$ are inverse functions by finding $f(g(x))$ and $g(f(x))$.

Hint: Start by putting $g(x)$ inside $f(x)$, like this: $2(\frac{x}{2} - 4) + 8$.

Answer: Yes, they are inverses because $f(g(x)) = x$ and $g(f(x)) = x$.

The Domain and Range Swap: A Closer Look

We learned earlier that when we find an inverse, the inputs ($x$) and outputs ($y$) trade places. This means the domain (all valid inputs) of the original function becomes the range (all valid outputs) of the inverse function.

Similarly, the range of the original function becomes the domain of the inverse!

Let's look closely at why this matters.

Consider the function $f(x) = x^2$. We learned this is not a one-to-one function because both $3^2 = 9$ and $(-3)^2 = 9$. It fails the Horizontal Line Test.

So, how can we possibly find an inverse for it?

Restricting the Domain

We can "rescue" $f(x) = x^2$ by cutting off half of its graph! If we only look at zero and positive numbers, we are restricting the domain to $x \geq 0$.

Now, the graph is just the right side of the U-shape. It passes the Horizontal Line Test!

For $f(x) = x^2$ where $x \geq 0$:

• The domain is $x \geq 0$ (or $[0, \infty)$).
• The range is $y \geq 0$ (or $[0, \infty)$).

Now let's find the inverse. The inverse of squaring a number is taking the square root. So, $f^{-1}(x) = \sqrt{x}$.

• The domain of the inverse is $x \geq 0$. (This matches the range of the original!)
• The range of the inverse is $y \geq 0$. (This matches the restricted domain of the original!)

Worked Example: Domain and Range

Question: The function $f(x) = \sqrt{x - 5}$ has a domain of $[5, \infty)$ and a range of $[0, \infty)$. Find the domain and range of its inverse function, $f^{-1}(x)$.

Step 1: Remember the golden rule of inverses: the domain and range swap places.

Step 2: Take the range of $f(x)$, which is $[0, \infty)$, and make it the domain of $f^{-1}(x)$.

Step 3: Take the domain of $f(x)$, which is $[5, \infty)$, and make it the range of $f^{-1}(x)$.

Answer: For $f^{-1}(x)$, the domain is $[0, \infty)$ and the range is $[5, \infty)$.

Your Turn

Question: A one-to-one function $h(x)$ has a domain of $(-\infty, 4)$ and a range of $(2, 10]$. What are the domain and range of $h^{-1}(x)$?

Hint: Just swap the sets!

Answer: The domain of $h^{-1}(x)$ is $(2, 10]$ and the range is $(-\infty, 4)$.

What is an Inverse Function?

Have you ever wondered how to flip a math function inside out?

Think of a function $f(x)$ as a mathematical machine. If you drop a $2$ into the machine, and it spits out a $5$, the function maps $2 \rightarrow 5$.

An inverse function, written as $f^{-1}(x)$, is simply a reverse machine. It takes that $5$, runs it backwards, and spits out the original $2$. So, $f^{-1}(x)$ maps $5 \rightarrow 2$.

Comparing Functions and Inverses

Let's compare a function and its inverse carefully:

• Inputs and Outputs swap: If the point $(x, y)$ is on the graph of $f(x)$, then the flipped point $(y, x)$ must be on the graph of $f^{-1}(x)$.
• Domain and Range swap: The domain (all possible $x$-values) of the original function becomes the range (all possible $y$-values) of the inverse. The range of the original function becomes the domain of the inverse!

The Line of Reflection

What does this swapping look like visually? If you plot a bunch of points $(x, y)$ and then plot their reversed twins $(y, x)$, you will notice they mirror each other perfectly across a diagonal line.

This magic mirror is the straight line $y = x$.

To graph the inverse of any function, you just reflect its graph across the line $y = x$.

Step-by-Step: Graphing an Inverse

1. Draw the Mirror: Lightly draw the diagonal line $y = x$ on your graph.
2. Pick Points: Find a few clear, easy-to-read points on the original graph of $f(x)$.
3. Swap Coordinates: For every point $(a, b)$, create a new point $(b, a)$.
4. Plot and Connect: Plot your new reversed points. Connect them smoothly to reveal the graph of $f^{-1}(x)$!

Notice how if you folded the graph paper along the gray dashed line $y = x$, the blue curve and the red curve would land exactly on top of each other! This perfectly visualizes why $f(x)$ and $f^{-1}(x)$ are inverses.