Inverse Functions explained through a friendly whiteboard conversation, examples, visuals, and practice for school students.
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Notes
Step 2
Formulae
Step 3
Examples
Step 4
Practice
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An inverse function reverses the action of a function.
If a function changes x into y, its inverse changes y back into x.
Inverse functions are used in algebra, graphs, logarithms, trigonometry, and reaorld modelling.
A function gives one output for each input in its domain.
The inverse reverses input and output.
To find an inverse, write , swap x and y, then solve for y.
The domain of the function becomes the range of the inverse.
Not every function has an inverse that is also a function unless the original function is onne on the chosen domain.
Meaning: A function connects each input to one output.
Definition: f(x) is the output when input is x.
For , input 4 gives output 11.
Meaning: An inverse reverses a function.
Definition: If , then f inverse of b is a.
If f changes 2 into 7, the inverse changes 7 back into 2.
Meaning: Swap input and output, then solve.
Definition: Write , swap x and y, then isolate y.
This shows the reverse rule.
Meaning: Composition means putting one function inside another.
Definition: For inverse functions, f(f^-1(x)) = x and f^-1(f(x)) = x.
The two functions undo each other.
Easy
Question: Find the inverse of .
Medium
Question: Find the inverse of .
Exam-level
Question: Check whether and are inverses.
Reading f^-1(x) as reciprocal
Why it happens: The notation looks like a power.
Correct approach: f^-1 means inverse function, not .
Forgetting to swap x and y
Why it happens: Students solve the original function again.
Correct approach: Swap first, then solve for y.
Ignoring domain restrictions
Why it happens: The graph is not checked.
Correct approach: Check whether the inverse passes the function rule on the chosen domain.
Does every function have an inverse function?
Not always. The inverse must also give only one output for each input.
What does f^-1 mean?
It means the inverse function.
How do I check my answer?
Compose the function and inverse. The result should be x.
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