Finding inputs from an output
Distinguish evaluating a function from finding inputs meeting an output condition.
Begin with the activity ↓Finding inputs from an output
For f(x)=2x+1, input three gives output seven. Now ask, “Which input gives seven?” The relationship is the same, but the unknown has moved to the input side.
For f(x)=2x+1, input three gives output seven. Now ask, “Which input gives seven?” The relationship is the same, but the unknown has moved to the input side.
With domain integers 0,1,2,3,4, find seven among outputs 1,3,5,7,9. Its row has input three. Distinguish calculating f(3) from finding x satisfying f(x)=7.
An equality such as f(x)=7 can be viewed as an equation asking for inputs meeting an output condition. You do not need a symbolic rearrangement rule yet. First find every matching row and verify it in the expression.
For the constant function f(x)=3, all five inputs give three. No input gives four. A function guarantees one output for each input, not one input for each output when read backward.
Even for f(x)=2x+1, output four is absent on this integer domain. Solving the expression gives x=1.5, but it is not an allowed input here. Changing the domain to a real interval can change the answer.
Change the rule and target output below, finding all matching inputs. One, several or none are all possible. A later course can connect equations with how outputs change as inputs increase.
Listen one paragraph at a time
Hold on to this distinction
Reversing a function’s arrows does not automatically produce another function. Find all inputs reaching the target and exclude values outside the domain.
Change it and check
Predict how many inputs match the target. Choose a rule and target, then read every marked table row.
Domain: 0, 1, 2, 3, 4 · f(x) = 2x + 1
| x | f(x) |
|---|---|
| 0 | 1 |
| 1 | 3 |
| 2 | 5 |
| → 3 | 7 |
| 4 | 9 |
f(x) = 7
Matching inputs: 3
Count: 1
An output can have one, several or no matching inputs.
The domain is integers 0,1,2,3,4. Targets range from zero through nine. A codomain value outside the range has no matching input. Formal inverse-function notation and equation transformations remain later topics.
Try with paper and objects
Circle the row with output seven in 0→1, 1→3, 2→5, 3→7, 4→9. Make another table with every output three, then search for targets three and four.
Check and explain another example
Check your idea 1
Check your idea 2
Apply it in a new setting
Write an answer yourself
Return to the explanationCompare with a solution
4 · input value
2×4+1=9, and four is allowed.
Does your explanation include these ideas?
Explain aloud or in a drawing, then check your ideas. Different wording can express the same meaning.