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By the end of these notes, you should be able to:
Think of a function like a machine. You put a number in, the machine does something to it, and out comes a new number.
An inverse function is a second machine that undoes exactly what the first machine did. If you feed the output of the first machine into the inverse machine, you get back the original number you started with.
Example: If f(x) = x + 3, then putting in x = 5 gives you 8. The inverse function would take 8 and give you back 5 — it undoes the "+3" by doing "−3".
The inverse of a function f(x) is written as f⁻¹(x).
⚠️ Important notation warning: The "−1" in f⁻¹(x) is not a power. It does not mean 1/f(x). It is simply the special symbol used to show an inverse function.
Not every function has an inverse. There is one very important rule:
A function can only have an inverse if it is a one–one function (also called a one–one mapping).
What is a one–one function? A function is one–one if every input (x-value) gives a different output (y-value). No two different inputs produce the same output.
Think of it this way: if you reverse the machine, each output must point back to exactly one specific input. If two inputs gave the same output, the reversed machine wouldn't know which original input to go back to — so the inverse would not be a proper function.
Example of a one–one function: f(x) = 5x − 2 is one–one because every different x gives a different answer. ✅
Example of a function that is NOT one–one: f(x) = x² (for all real x) is not one–one because, for example:
Both x = 3 and x = −3 give the same output of 9. If you tried to reverse this, you'd ask: "which x gave me 9 — was it 3 or −3?" You can't tell. So f(x) = x² does not have an inverse when defined for all real numbers.
💡 Fix: You can sometimes restrict the domain (the set of allowed x-values) to make a function one–one. For example, if you only allow x ≥ 0 for f(x) = x², then every output comes from exactly one input, and an inverse can exist.
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