DPST1013 Chap.8 Inverse Functions and Curve Sketching
Inverse Functions and Curve Sketching
An inverse function answers a reconstruction question: which permitted input produced this output? That question has a unique answer only when the original function is one-to-one. Restricting the domain can make an otherwise ambiguous rule invertible, but the restriction must remain visible throughout the calculation.
When solving an equation introduces two square-root branches, the original domain selects the admissible branch. The inverse exchanges the original domain and range, so its own allowed inputs are determined by outputs that were actually attained. Curve sketching combines this attention to domains with limits and derivatives. Begin with restrictions, intercepts and symmetries.
Then use limits to identify vertical, horizontal or oblique asymptotes. Derivative signs explain increasing and decreasing intervals, while second derivatives describe concavity. A plotted curve should communicate those calculated facts. It should never connect across an excluded input merely because a drawing tool joins adjacent samples.
Parametric and polar descriptions extend the same reasoning to curves whose coordinates depend on another variable. Parametric equations retain direction of travel and can revisit the same point, so a tangent question may require identifying more than one parameter value. Polar equations describe radius as a function of angle; the Cartesian curve is obtained by calculating both coordinates.
Negative radii and repeated angular sweeps need deliberate interpretation. Throughout the chapter, an exact landmark or derivative calculation is stronger evidence than an unlabelled picture. The final goal is a justified geometric account of where the curve exists, how it moves, and what local or limiting features it has.
What this chapter covers
- 01
One-to-one functions: test whether two different allowed inputs can share an output. A horizontal-line interpretation is useful, but a strict monotonicity argument can provide a precise proof. Any chosen restriction becomes part of the function definition.
- 02
Inverse construction: solve the output equation for the input, enforce the original restriction, and exchange domain with range. Check by composing in both directions on the appropriate sets. An inverse operation is different from taking a reciprocal.
- 03
Inverse derivatives and trigonometric branches: the derivative of the inverse is the reciprocal of the original derivative at the corresponding input, when the derivative is nonzero. Principal inverse trigonometric ranges determine which angles are returned.
- 04
Cartesian features: establish asymptotes with limits and turning points with derivative sign changes. A zero second derivative alone does not prove an inflection, and an excluded input cannot become a point of the curve.
- 05
Parametric and polar geometry: differentiate the coordinate pair and divide the vertical rate by the horizontal rate when permitted. In polar form use x=r cosθ and y=r sinθ, retaining the specified angle interval and direction of tracing.
An inverse on a decreasing branch
- 2On x≤3 the quantity x−3 is nonpositive. The squared term decreases to zero as x approaches three, so f is one-to-one on this branch and its attained outputs are all numbers at least two. Those outputs form the inverse domain.
- 2Writing y−2=(x−3)² gives x−3=−√(y−2), because the nonpositive branch is required. Thus the inverse rule is g(y)=3−√(y−2), for y≥2. The positive square root branch would return inputs outside the specified original domain.
- 2Differentiate to obtain g′(y)=−1/(2√(y−2)) for y>2. At output eleven, the square root is three, giving slope −1/6. Independently, g(11)=0 and f′(0)=−6, whose reciprocal agrees with the inverse slope.
Key terms
- Injective function
- A function that never assigns the same output to two different inputs in its stated domain; this property permits an inverse mapping.
- Principal branch
- A specified restriction which makes a many-to-one rule one-to-one and fixes the output range of its standard inverse.
- Oblique asymptote
- A sloping line whose vertical difference from a graph tends to zero along a specified infinite direction.
- Parameter
- An auxiliary variable which determines both coordinates of a curve and can preserve direction or repeated traversal information.
- Polar radius
- The signed radial coordinate associated with an angle; a negative value places the point in the opposite direction.
- Vertical tangent
- A local tangent aligned with the vertical axis, requiring attention when the horizontal coordinate rate vanishes.
Inverse Functions and Curve Sketching FAQ
Why is an inverse different from a reciprocal?
The inverse undoes a mapping and returns an input from an output. A reciprocal instead divides one by the output. They have different rules, domains and purposes, even when notation looks superficially similar.
Can a function have several inverse branches?
A many-to-one rule can become invertible on different restricted domains. Each restriction gives its own inverse branch, so there is no unique unrestricted inverse unless the original rule is already one-to-one.
Does a concavity change across an asymptote give an inflection?
No point of the graph exists at an excluded vertical asymptote. Concavity can differ on disconnected branches without producing an inflection point. Establish the relevant point and a genuine local concavity change.
What if both parametric derivatives are zero?
The usual ratio cannot be evaluated by dividing zero by zero. Reconsider the parameterisation, inspect nearby slopes or use a limiting argument. The vanishing rates alone do not identify a unique tangent.
Why must a polar graph be converted to coordinates?
A radius-versus-angle plot shows how the radial value changes, but its axes are not Cartesian position axes. Calculating x and y places each radius-angle pair at its correct position in the plane.
Exam move
Work in three passes. First practise inverse problems with explicit branch restrictions, and check every answer through substitution. Include increasing and decreasing branches so that selecting a square-root sign becomes a logical step rather than a habit. Second sketch one rational function using a written feature list before drawing: domain, intercepts, limits, derivative signs and concavity.
Mark excluded inputs prominently, and compare your finished picture with each calculation. Finally practise one parametric and one polar tangent question. Record the parameter, point, two coordinate derivatives and resulting slope in separate lines. Check the horizontal derivative before dividing. For revision, explain aloud why each restriction is needed.
An explanation of a failed hypothesis is often more valuable than repeating a successful calculation, because it prepares you to distinguish similar-looking questions that require different branches or methods.
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