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2.8 · Inverse Concept
What does an inverse relation do? How is it found from a table?
Think: what does it "undo"?
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2.8 · Inverse Notation
How is the inverse of f(x) written? What does it mean if (a, b) is on f?
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2.8 · Graphical Property
What is the graphical relationship between a function and its inverse?
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2.8 · Steps: Find Inverse Equation
What are the steps to find f ⁻¹(x) from an equation?
y = f(x) → switch x & y → ?
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2.8 · Apply: Linear Inverse
Find f ⁻¹(x) for f(x) = 3x − 7.
Switch x and y, then solve for y
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2.8 · Apply: Rational Inverse
Find f ⁻¹(x) for f(x) = (2x+1)/(x−3).
Switch, multiply by denominator, collect y-terms, factor, divide
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2.8 · Verify Inverses
What must you show to verify that f and g are inverse functions?
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2.8 · Apply: Verify
Show that f(x) = 2x−3 and g(x) = ½x + 3/2 are inverses.
Compute both f(g(x)) and g(f(x))
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✓ Inverse Notation
f ⁻¹(x) — if (a, b) is on f, then (b, a) is on f ⁻¹
Domain of f = Range of f⁻¹. Range of f = Domain of f⁻¹.
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✓ Inverse Concept
It "undoes" the original. Found by switching every x and y value.
The inverse may not be a function if any x-value in the new table repeats.
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✓ Steps: Find Inverse
1. Replace f(x) with y 2. Switch x and y 3. Solve for y 4. Write as f⁻¹(x)
For rational functions: also clear fractions, collect y-terms, factor, and divide.
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✓ Graphical Property
Inverses are reflections over the line y = x
Switch x and y coordinates of key points to find the inverse. Points already on y = x stay fixed.
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✓ Rational Inverse
f ⁻¹(x) = (3x+1)/(x−2)
x(y−3)=2y+1 → xy−3x=2y+1 → xy−2y=3x+1 → y(x−2)=3x+1 → y=(3x+1)/(x−2).
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✓ Linear Inverse
f ⁻¹(x) = (x+7)/3
Switch: x = 3y−7. Add 7: x+7 = 3y. Divide: y = (x+7)/3.
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✓ Apply: Verify
f(g(x)) = (x+3)−3 = x ✓ g(f(x)) = (x−3/2)+3/2 = x ✓
Both compositions equal x, confirming f and g are inverses.
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✓ Verify Inverses
Show BOTH f(g(x)) = x AND g(f(x)) = x
One direction is not enough! You must verify both compositions equal x to prove they are inverses.
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2.8 · Key Properties
What is the relationship between the domain and range of f and f ⁻¹?
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2.8 · Inverse from Graph
f is defined on [−2, 8] with range [−3, 7]. What is the domain of f ⁻¹?
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2.8 · Reading f ⁻¹ from Graph
How do you evaluate f ⁻¹(a) using the graph of f?
What are you really looking for?
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2.8 · Inverse from Table
f table: f(−3)=6, f(−2)=3, f(0)=1, f(1)=−1, f(4)=−3, f(6)=−7. What is f ⁻¹(−3)?
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✓ Domain of f ⁻¹
Domain of f ⁻¹ = [−3, 7]
Domain of f⁻¹ = Range of f = [−3, 7]. Range of f⁻¹ = Domain of f = [−2, 8].
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✓ Domain & Range Switch
Domain of f = Range of f ⁻¹ | Range of f = Domain of f ⁻¹
Because inverses swap x and y, all domain and range information is also swapped.
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✓ Inverse from Table
f ⁻¹(−3) = 4
Find the x-value where f(x) = −3. From the table, f(4) = −3. So f⁻¹(−3) = 4.
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✓ Reading f ⁻¹ from Graph
Find the x-value where f(x) = a (reverse lookup)
f⁻¹(6) = 11 means "what x gives f(x) = 6?" Look at the graph and find where y = 6, then read the x-coordinate.