DPST1013 Chap.8 Inverse Functions and Curve Sketching
Inverse Functions and Curve Sketching
Inverse functions reverse one-to-one mappings and reflect graphs across the line y equals x. Domain restrictions determine whether an inverse exists and which algebraic branch is valid. Curve sketching combines domain, intercepts, limits, asymptotes, derivatives, concavity and parameter behaviour into an evidence-based picture.
What this chapter covers
- 01
One-to-one functions
- 02
Domain and range exchange
- 03
Inverse derivatives
- 04
Asymptotes and intercepts
- 05
Monotonicity and concavity
- 06
Parametric and polar curve tracing
Worked example · free
Invert a restricted quadratic branch
- 1The restricted branch is increasing and therefore one-to-one.
- 1Solve y - 1 = (x - 2) squared with x - 2 non-negative.
- 1Choose x = 2 + sqrt(y - 1), then swap variable labels.
- 1The inverse is 2 + sqrt(x - 1) on x at least 1; composition verifies it.
Key terms
- Inverse function
- A function reversing a one-to-one mapping by exchanging inputs and outputs.
- Horizontal-line test
- A graphical test requiring each horizontal line to meet a one-to-one graph at most once.
- Vertical asymptote
- A finite x-value near which the function becomes unbounded.
- Inflection point
- A point on the graph where concavity changes, subject to domain and continuity checks.
Inverse Functions and Curve Sketching FAQ
Why must a domain be restricted before inversion?
A function such as a full quadratic sends two inputs to the same output, so reversing it would not produce a function. Restricting to one monotone branch creates a one-to-one mapping and fixes the square-root sign.
Where is the inverse derivative evaluated?
If f(a) equals b, then the inverse derivative is evaluated at b and equals one divided by f prime(a), provided that derivative is non-zero. The paired points swap coordinates on the reflected graphs.
What evidence belongs on a curve sketch?
Label the domain, intercepts, asymptotes, extrema and inflection behaviour supported by limits and derivatives. A smooth drawing without those calculations is a guess, while a labelled sketch summarises the analysis.
How do domain and range change under inversion?
The inverse swaps the roles of input and output, so its domain is the original function's range and its range is the original domain. This exchange also swaps every graph point from ordered pair a, b to b, a. State any original restriction before solving the inverse formula, because that restriction determines which branch and sign make the reversed relation a function.
What is a reliable algebraic check for an inverse formula?
Compose in both directions on the correct domains. Substituting the inverse into the original should return the inverse's input, while substituting the original into the inverse should return the original input. Simplifications involving squares or roots require special care because an absolute value or restricted sign may appear. A single numerical test is useful but cannot replace the domain-aware composition argument.
How do derivative sign charts support a curve sketch?
Mark critical numbers and domain breaks on a number line, test the derivative's sign in every interval, and translate positive or negative signs into increasing or decreasing behaviour. A change from positive to negative identifies a local maximum; the reverse identifies a local minimum. Do not connect across excluded points, and use endpoint or limit information to decide how each branch approaches its boundary.
How should asymptotes be justified rather than guessed?
A vertical asymptote follows from an infinite one-sided limit at a finite input. A horizontal asymptote follows from a finite limit as the input tends to positive or negative infinity, and the two ends may differ. An oblique asymptote requires the difference between the function and a line to tend to zero. State the relevant limit beside each claimed asymptote.
What extra care is needed for parametric curves?
Treat both coordinates as functions of the same parameter and track the allowed parameter interval. Eliminate the parameter only when it does not lose orientation or duplicate branches. The slope is the derivative of y with respect to the parameter divided by the derivative of x, where the denominator is non-zero. Mark starting points, endpoints and traversal direction on the sketch.
Exam move
State the original domain and range before solving for an inverse. Use monotonicity or the horizontal-line test to justify one-to-one behaviour, then check both compositions on their proper domains. For curve sketches, follow a fixed evidence order: domain, symmetry, intercepts, limits, asymptotes, first derivative and second derivative.
For polar or parametric curves, track the tracing parameter and test symmetry algebraically rather than relying only on appearance. Divide a page into algebraic evidence and graphical consequences, connecting each limit or sign result to one visible feature. When restricting a domain, explain why the chosen interval is one-to-one and what range it produces.
Practise reflecting a few labelled points across the diagonal before deriving an inverse formula. On full sketch questions, make separate sign charts for first and second derivatives, then reconcile them with endpoint behaviour rather than drawing from memory.
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