Lesson 3 of 4
Recover an input while preserving the original model’s range and units.
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Big ideas · 1 of 8
Big ideas
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A linear function with is one-to-one. Its inverse rule is , with a domain taken from the original range.
The original rule multiplies then adds. The inverse subtracts the intercept then divides by the slope; parentheses keep the whole difference in the numerator.
If the original maps hours to dollars, its inverse maps dollars to hours. The inverse slope has reciprocal units as well as a reciprocal numerical value.
A restricted original domain gives a restricted inverse range. Only original possible outputs can be valid inverse inputs, even if the inverse formula accepts more real numbers algebraically.
Step-through
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Worked example: reverse a cost model
Whole picture, 4 steps
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Check .
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The endpoints trade roles without reversing interval order.
Do this: Explain why the tempting shortcut fails.
Wrong idea: the inverse of is . Undo the last operation first: subtract 6, then divide the entire difference by 3. The correct inverse is .
Worked example
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Worked example
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Solve for input to get on .
Watch
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Activity 1 of 7 · 0 of 7 activities done
Flashcards
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Card 1 of 8
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Undo addition before multiplication.
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What is the inverse of mx+b when ?
.
Why can a constant linear rule not have an inverse on a multi-input domain?
Its outputs repeat, so it is not one-to-one.
What is the inverse of ?
.
If a model maps minutes to meters, what does its inverse map?
Meters to minutes.
What is the inverse slope of a line of slope −4?
.
What is the inverse domain if the original range is ?
.
How do you check a proposed linear inverse?
Compose it with the original and check that allowed inputs return.
Why state units in an inverse model?
The answer describes a different quantity than the original output.
True or false
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Self-test
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6 questions. One at a time.
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Self-test
Do this: Try each problem without a timer. Read the feedback before moving on.
6 questions. One at a time.
Practice, not a grade. No timer. Your best score stays in this browser.
Write it
Do this: Write two or three sentences, then compare with the model.
Start here if it helps: I would first identify...
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Write it
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Match
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Connect the ideas
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Homework · 3 points
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Homework
Worth 3 points.
https://gradebridge-beryl.vercel.app/learn/precalculus/linear-functions/linear-inverses#homework
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