MATH1131 Chap.10 Inverse Functions and Curve Sketching
Inverse Functions and Curve Sketching
A function has an inverse exactly when it is one-to-one, and graphically the inverse is the reflection of the graph in the line y = x, with domain and range exchanged.
On an interval, a continuous function with a derivative of constant sign is strictly monotonic and therefore one-to-one, and the inverse function theorem states that the inverse is differentiable with derivative equal to the reciprocal of the original derivative at the matching point.
Restricting sine, cosine and tangent to suitable intervals produces the inverse trigonometric functions, whose derivatives follow from that theorem, and whose restricted ranges matter when simplifying compositions.
The second half of the chapter assembles curve sketches from calculations: domain, intercepts, symmetry, vertical, horizontal and oblique asymptotes, stationary points and their nature, and behaviour near asymptotes.
The same approach extends to curves defined parametrically, where slopes come from the ratio of the two time derivatives, and to curves in polar coordinates, which are converted using x = r cos theta and y = r sin theta.
What this chapter covers
- 01
One-to-one functions and the horizontal-line test
- 02
Inverse functions and reflection in y = x
- 03
The inverse function theorem
- 04
Inverse trigonometric functions and their derivatives
- 05
Asymptotes of rational functions
- 06
Stationary points and the shape of a graph
- 07
Parametric curves
- 08
Polar coordinates and polar curves
Worked example · free
Derivative of an inverse at a point
- 1f'(x) = 5x^4 + 2 > 0 for all x, so f is strictly increasing and therefore one-to-one.
- 1Find the matching input: f(1) = 1 + 2 + 1 = 4, so the inverse sends 4 to 1.
- 1Inverse function theorem: the derivative of the inverse at 4 is 1/f'(1).
- 1f'(1) = 5 + 2 = 7, so the derivative of the inverse at 4 is 1/7.
Key terms
- One-to-one function
- A function that never sends two different inputs to the same output.
- Inverse function
- The function that reverses a one-to-one function, sending each output back to its input.
- Oblique asymptote
- A slanted line that a graph approaches as x tends to plus or minus infinity.
- Polar coordinates
- A description of a point by its distance r from the origin and its angle theta from the positive x-axis.
Inverse Functions and Curve Sketching FAQ
Why must sine be restricted before it has an inverse?
Sine repeats its values, so it is not one-to-one on the real line. Restricting it to the interval from minus pi over 2 to pi over 2, where it is increasing, makes it one-to-one and defines the inverse sine.
How do I find the derivative of an inverse function at a point?
Find the input x that the original function sends to the given point, compute the original derivative at that x, and take its reciprocal. This requires the original derivative to be non-zero there.
How do I find an oblique asymptote of a rational function?
When the numerator degree is one more than the denominator degree, divide the polynomials. The quotient gives the asymptote, because the remainder term tends to zero as x grows.
How do I convert a polar equation to Cartesian form?
Use x = r cos theta, y = r sin theta and r squared = x squared + y squared. Multiplying the polar equation by r first often helps, as with r = 2 cos theta becoming x squared + y squared = 2x.
What must a curve sketch show to earn full marks?
All computed features labelled on the graph: intercepts, asymptotes with their equations, and stationary points with coordinates, together with behaviour that matches the sign of the derivative between them.
How do I find the slope of a parametric curve?
Differentiate both coordinates with respect to the parameter and divide: dy/dx equals dy/dt divided by dx/dt, wherever dx/dt is not zero. Horizontal tangents occur where dy/dt is zero and vertical ones where dx/dt is zero.
Exam move
For inverse functions, practise the three-step pattern of proving monotonicity, locating the matching input, and taking a reciprocal derivative. Memorise the derivatives of the inverse trigonometric functions and practise deriving one of them from the inverse function theorem, since that derivation is a typical written question. For sketching, use a fixed checklist and fill it in completely before drawing anything.
Compare your sketches with a plot in Maple afterwards to see which features you missed. Practise reading features back off a finished sketch, such as where the function increases and which values it never takes, because questions sometimes ask for the range after the sketch.
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