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Inverse Functions (linear Skill)
If f is bijective, f^{-1} undoes f. For a linear f, set y=f(x), solve for x, then swap the letter so the inverse is written as a function of x. Check: f(f^{-1}(x))=x and f^{-1}(f(x))=x. Inverse is not 1f(x) unless the paper is talking about reciprocal functions.
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Inverse Functions (linear Skill)
What is it?
If f is bijective, f^{-1} undoes f. For a linear f, set y=f(x), solve for x, then swap the letter so the inverse is written as a function of x. Check: f(f^{-1}(x))=x and f^{-1}(f(x))=x. Inverse is not 1f(x) unless the paper is talking about reciprocal functions.
In simple words
- If f is bijective, f-1 undoes f.
- FINDING IT: y=2x-1 -> x=y+12 -> f-1(x)=x+12.
- CHECK: f(f-1(5))=f(3)=5.
- Worked-method reminder: check works
Important terms
INVERSE FUNCTIONS (LINEAR SKILL): If f is bijective, f^{-1} undoes f. For a linear f, set y=f(x), solve for x, then swap the letter so the inverse is written as a function of x. Check: f(f^{-1}(x))=x and f^{-1}(f(x))=x. Inverse is not 1f(x) unless the paper is talking about reciprocal functions.
FINDING IT: y=2x-1 -> x=y+12 -> f-1(x)=x+12. y=(12)x -> f-1(x)=2x. The graph of f-1 is the reflection of y=f(x) in y=x.
CHECK: f(f-1(5))=f(3)=5. If the check fails, the algebra of the inverse failed. Domain of f-1 is the range of f.
What you should learn
- Explain inverse functions (linear skill) using the correct definitions, conditions, and notation.
- Apply solve for x, f-1 is not 1f correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Examples
- x+12
- 2x for half-x
- Not 12x-1
FINDING IT
y=2x-1 -> x=y+12 -> f-1(x)=x+12. y=(12)x -> f-1(x)=2x. The graph of f-1 is the reflection of y=f(x) in y=x.
CHECK
f(f-1(5))=f(3)=5. If the check fails, the algebra of the inverse failed. Domain of f-1 is the range of f.
Recommended for Class 10
Full school-level explanation
If f is bijective, f^{-1} undoes f. For a linear f, set y=f(x), solve for x, then swap the letter so the inverse is written as a function of x. Check: f(f^{-1}(x))=x and f^{-1}(f(x))=x. Inverse is not 1f(x) unless the paper is talking about reciprocal functions.
Undo the linear rule:
WHAT — INVERSE FUNCTIONS (LINEAR SKILL)
If f is bijective, f^{-1} undoes f. For a linear f, set y=f(x), solve for x, then swap the letter so the inverse is written as a function of x. Check: f(f^{-1}(x))=x and f^{-1}(f(x))=x. Inverse is not 1f(x) unless the paper is talking about reciprocal functions. The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Inverse Functions (Linear Skill) matters because functions describe how one quantity depends on another. Domain, range, composition, and inverse ideas are the language used for graphs and mathematical models.
HOW — How to solve and explain it
Start by identifying what is given and what must be found. Work through FINDING IT, then CHECK. Worked: f(x) = (12)x. Find f-1(x) and check at x = 6. y = (12)x 2y = x f-1(x) = 2x f-1(6) = 12 f(12) = (12)*12 = 6 check works Finish by checking signs, restrictions, units, and whether the answer matches the question.
FINDING IT
y=2x-1 -> x=y+12 -> f-1(x)=x+12. y=(12)x -> f-1(x)=2x. The graph of f-1 is the reflection of y=f(x) in y=x.
Remember
- reflect in y=x
Examples
- x+12
- 2x for half-x
- Not 12x-1
CHECK
f(f-1(5))=f(3)=5. If the check fails, the algebra of the inverse failed. Domain of f-1 is the range of f.
Remember
- compose both ways
Examples
- f(f-1(x))=x
- Range becomes domain
- Need bijection
Remember
- solve for x
- f-1 is not 1f
- f-1(x)=2x
- inverse vs reciprocal
- reflect in y=x
- compose both ways
- solve for x
- f-1 is not 1f
How the ideas connect
Worked: f(x) = (12)x. Find f-1(x) and check at x = 6. y = (12)x 2y = x f-1(x) = 2x f-1(6) = 12 f(12) = (12)*12 = 6 check works
Deeper understanding
Go beyond memorising Inverse Functions (linear Skill). Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: f(x) = (12)x. Find f-1(x) and check at x = 6. y = (12)x 2y = x f-1(x) = 2x f-1(6) = 12 f(12) = (12)*12 = 6 check works
Common mistakes
Exam focus
Core Idea
FINDING IT
CHECK
Worked Method
Exam focus
Quick revision
- solve for x
- f-1 is not 1f
- f-1(x)=2x
- inverse vs reciprocal
- INVERSE FUNCTIONS (LINEAR SKILL): If f is bijective, f-1 undoes f. For a linear f, set y=f(x), solve for x, then swap the letter so
- FINDING IT: y=2x-1 -> x=y+12 -> f-1(x)=x+12. y=(12)x -> f-1(x)=2x. The graph of f-1 is the refle
f-1(x) is inverse function notation. The -1 is not an exponent here. 1f(x) would be written [f(x)]-1 or 1f(x).
For long answers on Inverse Functions (linear Skill): define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- solve for x
- f-1 is not 1f
- f-1(x)=2x
- inverse vs reciprocal
Important formulas
- solve for x
- f-1 is not 1f
- f-1(x)=2x
Common exam traps
- f-1(x) is inverse function notation. The -1 is not an exponent here. 1f(x) would be written [f(x)]-1 or 1f(x).
- Not 12x-1
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
f-1(x) is inverse function notation. The -1 is not an exponent here. 1f(x) would be written [f(x)]-1 or 1f(x).
- Mistake
Not 12x-1
- Mistake
Do not treat FINDING IT and CHECK as the same step; each has a different purpose.
- Mistake
f-1(x) is inverse function notation. The -1 is not an exponent here. 1f(x) would be written [f(x)]-1 or 1f(x).
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
solve for x
f-1 is not 1f
f-1(x)=2x
inverse vs reciprocal
If f is bijective, f-1 undoes f. For a linear f, set y=f(x), solve for x, then swap the letter so the inverse is written as a function of x. Check: f(f-1(
f-1(x)=2x
- Definition
- If f is bijective, f-1 undoes f. For a linear f, set y=f(x), solve for x, then swap the letter so the inverse is written as a function of x. Check: f(f-1(
- Formula
- f-1(x)=2x
- Example
- x+12
- Critical warning
- f-1(x) is inverse function notation. The -1 is not an exponent here. 1f(x) would be written [f(x)]-1 or 1f(x).
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Exam Practice
Define
Q1. Define Inverse Functions (linear Skill).
Explain
Q2. Explain: FINDING IT.
Example
Q3. Give one clear example of Inverse Functions (linear Skill) and say why it fits.
Mistake
Q4. State one common mistake students make about Inverse Functions (linear Skill).
Exam note
Q5. Write a short exam note on Inverse Functions (linear Skill): definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.