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Topic 2.8 · Core Concept
How do you find the inverse of any relation?
Works for tables, graphs, and equations
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Topic 2.8 · Notation
What does f⁻¹(x) mean? What does it NOT mean?
f⁻¹(x)
The −1 is NOT an exponent
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Topic 2.8 · Graphical Property
What is the graphical relationship between f and f⁻¹?
Reflect over which line?
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Topic 2.8 · Domain / Range Swap
Domain of f = [a,b], Range of f = [c,d]. What are domain and range of f⁻¹?
Everything swaps
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Topic 2.8 · Algebraic Method
List the steps to find f⁻¹(x) analytically from f(x) = (2x+1)/(x−3).
f(x) = (2x+1)/(x−3)
Switch x and y, then isolate y
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Topic 2.8 · Verification
What must you show to prove f and g are inverse functions?
f(g(x)) = ? and g(f(x)) = ?
BOTH compositions required
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Topic 2.8 · Table Evaluation
f(0)=1, f(1)=−1, f(4)=−3. Find f(f(0)) and f⁻¹(−3).
Work from inside out
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Topic 2.8 · Invertibility
A continuous function has an inverse function when it is…
Think horizontal line test
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✓ Notation f⁻¹(x)
Inverse function of f
f⁻¹(x) means the function that undoes f.
⚠️ f⁻¹(x) ≠ 1/f(x). The −1 means inverse, not reciprocal.
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✓ Core Concept
Switch every x and y value
Tables: swap rows. Equations: swap x and y, solve for y. Graph: reflect over y = x.
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✓ Domain/Range Swap
Domain f⁻¹=[c,d] · Range f⁻¹=[a,b]
Domain of f = Range of f⁻¹
Range of f = Domain of f⁻¹
Everything swaps when you invert.
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✓ Graphical Property
Reflection over y = x
Every (a, b) on f becomes (b, a) on f⁻¹. Points already on y = x stay fixed.
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✓ Verification
BOTH f(g(x))=x AND g(f(x))=x
Must show both compositions equal x. One alone is insufficient on the AP exam.
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✓ Inverse of (2x+1)/(x−3)
f⁻¹(x) = (3x+1)/(x−2)
Switch→x(y−3)=2y+1→xy−2y=3x+1→y(x−2)=3x+1→f⁻¹=(3x+1)/(x−2)
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✓ Invertibility
Strictly increasing OR decreasing
Must pass the horizontal line test. If f changes direction, its inverse fails the vertical line test.
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✓ Table Evaluation
f(f(0))=−1 · f⁻¹(−3)=4
f(0)=1, then f(1)=−1.
f⁻¹(−3): find x where f(x)=−3 → f(4)=−3 → f⁻¹(−3)=4.
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Topic 2.8 · Example 4 — Combined
g graph: g(10)=−1. h table: h(0)=3, h(6)=10, h⁻¹(10)=6. Find h⁻¹(g⁻¹(−1)).
g⁻¹(−1) = ? first
Work from inside out
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Topic 2.8 · Key Property
If (3, 7) is on f, what point is on f⁻¹? What does this mean for the graph?
Coordinates swap
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✓ Key Property: Point Swap
(7, 3) is on f⁻¹
Every (a,b) on f becomes (b,a) on f⁻¹. Graphically, the point is reflected over y=x.
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✓ Combined Composition
h⁻¹(g⁻¹(−1)) = 6
g⁻¹(−1)=10 (since g(10)=−1).
Then h⁻¹(10)=6 (since h(6)=10).