A graph point (3,4) says input 3 produces output 4. If the function is one-to-one, the inverse says input 4 returns 3. You can read that information without finding an algebraic rule. Inverse questions ask which original input produced a given output, so scan horizontally from the requested output before reading the original input.
Predict
Predict what happens when one output occurs at two original inputs. Reversing the relation gives two outputs for one inverse input, so it is not an inverse function. Predict whether a finite table defines values between its listed inputs. Unless an extension is specified, it does not; inverse lookup must respect that same limitation.
Experiment
Graph I. Exact increasing line segments join (−2,1),(0,3),(3,4),(5,7), including the endpoints.
Supplied Graph I is a complete piecewise-linear increasing function joining the listed vertices, including both endpoints. The separate finite table T has only its four listed inputs.
Graph I input
Graph I output
−2
1
0
3
3
4
5
7
Table T input
Table T output
−3
8
1
−2
4
5
9
11
Use the graph for one inverse lookup and the table for another. Explain why interpolation is justified for Graph I’s segments but not for the finite table.
Understand
A function f has an inverse function on its range when distinct inputs have distinct outputs. This property is called one-to-one. Graphically, every horizontal line meets a one-to-one graph at most once. The usual vertical-line test checks whether the original relation is a function; the horizontal-line test checks whether reversing it also gives a function.
An inverse graph is reflected across y=x because each coordinate pair (x,y) becomes (y,x). Its domain is the original range, and its range is the original domain. This is a swap of roles, not a reciprocal of the output values. In particular f inverse notation does not mean 1/f.
Graph I increases strictly along all its segments, so it is one-to-one on its complete interval domain. Swapping its vertices gives the inverse vertices, and a straight segment reflects to a straight segment. The inverse can be read or constructed even when the original is not presented by one simple formula. If both axes represent different physical quantities, the reflected coordinate roles also swap their units.
A full quadratic is generally not one-to-one: points on opposite sides of its vertex share outputs. Restricting its domain to one monotonic branch can produce an inverse. The restriction must be stated before selecting a square-root sign. Different branches can give different valid inverse functions, each for a different restricted original function.
For a finite table, check whether any outputs repeat. If none do, reverse every pair and retain only those output values as the inverse domain. An unlisted value is not automatically a request to estimate between points; that would require a continuous or other specified model beyond the table.
Apply
Give the inverse of Graph I as a list of vertices and joining segments. Then explain how to restrict a parabola so its inverse passes the vertical-line test. A graph-construction rubric should check reflected landmarks, domain/range, endpoint inclusion and the branch choice, not just one successfully reversed coordinate.
Continue
Inverse lookup is a representation skill as well as an algebra skill. It requires a one-to-one relationship and a declared domain. Next, inspect symmetry: even and odd classifications also depend on the domain, not merely on the appearance of a formula or a few plotted points.
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
Using supplied Graph I, find I^(-1)(4).
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
The finite table T has input-output pairs (−3,8),(1,−2),(4,5),(9,11). Find T^(-1)(5).
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Construct the inverse of supplied Graph I as a coordinate/segment description. Give its domain and range.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For the finite table T, list the domain of T inverse and explain why T inverse at 6 cannot be inferred by interpolation.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
The full parabola y=(x−1)² has outputs 4 at inputs −1 and 3. Explain why its inverse relation is not a function and give a restriction that produces an inverse, including the inverse value at 4.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A one-to-one graph maps elapsed time in minutes to distance in meters. Explain the units of the inverse input, inverse output and inverse secant rate.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
