Inverse Functions Revision Notes for Grade 12 NSC Matric Mathematics

Inverse Functions

What is an inverse function?

An inverse function is a function that reverses the operation of another function. Think of it as "undoing" what the original function does.

More formally, if we have a function $f$ with domain $X$ , then $f^{-1}$ is its inverse function when $f^{-1}(f(x)) = x$ for every value of $x$ in the domain $X$ .

For a function to have an inverse, it must be a one-to-one relation. This means each input value produces exactly one output value, and each output value comes from exactly one input value. When a function has an inverse, we say the function is invertible.

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The concept of "undoing" is key to understanding inverse functions. Just like subtraction undoes addition, or division undoes multiplication, an inverse function undoes whatever operation the original function performs.

Understanding function notation

Before working with inverses, let's clarify the notation:

$y = f(x)$ indicates a function
$y\_1 = f(x\_1)$ means we substitute a specific $x\_1$ value into the function to get the corresponding $y\_1$ value
$f^{-1}(y) = x$ indicates the inverse function
$f^{-1}(y\_1) = x\_1$ means we substitute a specific $y\_1$ value into the inverse to return the specific $x\_1$ value

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Understanding this notation is crucial for working with inverse functions. The superscript $-1$ in $f^{-1}$ is not an exponent - it's simply the symbol we use to denote the inverse function.

How to find the inverse of a function

To determine the inverse $f^{-1}(x)$ of a given function $f(x)$ , follow these three steps:

Interchange x and y in the equation
Make y the subject of the equation
Express the new equation in function notation

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Worked Example: Finding an Inverse Function

Let's find the inverse of $f(x) = 3x - 5$ :

Step 1: Start with $y = 3x - 5$ , then interchange $x$ and $y$ $x = 3y - 5$

Step 2: Make $y$ the subject $x = 3y - 5$ $x + 5 = 3y$ $y = \frac{x + 5}{3}$

Step 3: Express in function notation $f^{-1}(x) = \frac{x + 5}{3}$

Verification: If $f(x) = 3x - 5$ and $f^{-1}(x) = \frac{x + 5}{3}$ , then: $f(f^{-1}(x)) = f\left(\frac{x + 5}{3}\right) = 3\left(\frac{x + 5}{3}\right) - 5 = (x + 5) - 5 = x$ ✓

Graphical representation of inverse functions

When we graph a function $f$ and its inverse function $f^{-1}$ , the two graphs are reflections of each other about the line y = x.

This diagram shows an exponential function (black curve) and its inverse (blue curve) reflected about the line $y = x$ (grey dashed line).

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Any point on the line $y = x$ has $x$ - and $y$ -coordinates with the same numerical value, such as $(-3, -3)$ and $(4, 4)$ . This is why interchanging the $x$ - and $y$ -values makes no difference when finding inverses.

Important distinction: inverse vs reciprocal

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Be very careful not to confuse the inverse of a function with the reciprocal of a function. These are completely different concepts.

Key differences:

Inverse function

Notation: $f^{-1}(x)$
$f(x)$ and $f^{-1}(x)$ are symmetrical about the line $y = x$
Example: If $g(x) = 5x$ , then $g^{-1}(x) = \frac{x}{5}$

Reciprocal function

Notation: $[f(x)]^{-1} = \frac{1}{f(x)}$
Relationship: $f(x) \times \frac{1}{f(x)} = 1$
Example: If $g(x) = 5x$ , then the reciprocal is $\frac{1}{g(x)} = \frac{1}{5x}$

Critical notation note: The superscript $-1$ in $f^{-1}$ is not an exponent. It is simply the notation for indicating the inverse of a function. Do not confuse this with exponents like $(1/2)^{-1}$ or $3 + x^{-1}$ .

When a function has no inverse

If the inverse is not a function, then it cannot be written in function notation. For example, the inverse of $f(x) = 3x^2$ cannot be written as $f^{-1}(x) = \pm\sqrt{\frac{x}{3}}$ because it is not a function (one input gives two outputs). We write the inverse as $y = \pm\sqrt{\frac{x}{3}}$ and conclude that $f$ is not invertible.

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This is a common issue with quadratic and other even-powered functions. They fail the horizontal line test, which means they're not one-to-one and therefore don't have inverse functions.

Exam tips

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Essential Exam Strategies:

Always check your inverse by verifying that $f(f^{-1}(x)) = x$
Remember the three-step method: swap, solve, express
Watch for the distinction between $f^{-1}$ (inverse) and $\frac{1}{f}$ (reciprocal)
For graphical questions, remember inverse functions are reflections across y = x
A function must be one-to-one to have an inverse function

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Key Points to Remember:

Inverse functions "undo" the operation of the original function
Use the three-step method: interchange x and y, make y the subject, express in function notation
Graphs of inverse functions are reflections about the line y = x
f^(-1) means inverse, not reciprocal - don't confuse with $\frac{1}{f(x)}$
Only one-to-one functions are invertible and have inverse functions

Inverse Functions (Grade 12 NSC Matric Mathematics): Revision Notes