MathematicsChapter 6: Functions and Graphs
Inverse Functions (linear Skill)
If f is bijective, f^{-1} undoes f.
Learning objectives
- 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.
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(x))=x and f-1(f(x))=x. Inverse is not 1f(x) unless the paper is talking about reciprocal functions.
Mathematical form
y = (12)x
y=f(x)
x = 6. y
Where / symbols
- 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 1/f(x) unless the paper is talking about reciprocal functions.
- FINDING IT
- y=2x-1 -> x=(y+1)/2 -> f^{-1}(x)=(x+1)/2. y=(1/2)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.
Overview
Undo the linear rule:
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.
Examples (x+1)/2 · 2x for half-x · Not 1/(2x-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.
Examples f(f^{-1}(x))=x · Range becomes domain · Need bijection
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.
Worked example
- 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
In simple words
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).
Key points
- solve for x
- f-1 is not 1f
- reflect in y=x
- compose both ways
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Inverse Functions (linear Skill)
Sticky Notes · Quick Revision
Core Idea
- 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.
Worked Method
- 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 Misconception
- 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).
FBISE Exam Check
- 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).
60-second revision
- 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
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