2WBB0Calculus I

2WBB0 · Topic 01

Functions, domains, inverses, and algebra

The prerequisite moves hidden inside almost every multiple-choice and integration question.

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What you need to be able to do

  • Determine domains and ranges from logarithms, roots, denominators, and compositions.
  • Use one-to-one restrictions and the inverse-derivative rule.
  • Complete the square, factor polynomials, and divide rational functions before integrating.
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1. Domain, range, and composition

A domain question is a constraint-intersection problem. Collect every restriction before solving: a denominator may not vanish, a square-root input must be nonnegative, and a logarithm input must be positive.

For a composition, checking only the inner formula is not enough. The inner value must also land inside the domain of the outer function. Range questions are easiest after identifying monotonicity and endpoint behaviour.

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Exam method

  • Write one inequality or exclusion for every risky operation.
  • Solve the restrictions together and keep open versus closed endpoints correct.
  • For f composed with g, first require x in the domain of g, then require g(x) in the domain of f.
  • For a range, evaluate reachable endpoint values and use monotonicity or a substitution.
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Domain, range, and composition: worked examples

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Domain, range, and compositionSelf-review

Worked example 1

Determine the domain of f(x)=ln⁡x−3−ln⁡x.f(x)=\ln x-\sqrt{3-\ln x}.
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2. Inverse functions and inverse derivatives

An inverse swaps inputs and outputs. Before using an inverse, make sure the function is one-to-one on the stated interval; a monotone function passes this test.

The inverse-derivative rule is a two-step exam routine: find the original input that produces the requested inverse input, then take the reciprocal of the original derivative there.

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Exam method

  • Translate an inverse value b into the equation f(a)=b.
  • Solve for a on the stated one-to-one interval.
  • Differentiate f and evaluate f′(a).
  • Return 1/f′(a); do not substitute b into f′ unless a=b by coincidence.
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Inverse functions and inverse derivatives: worked examples

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Inverse functions and inverse derivativesSelf-review

Worked example 1

Let f(x)=xexf(x)=xe^x on (−1,∞)(-1,\infty). Find (f−1)′(e).(f^{-1})'(e).
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3. Algebra before calculus

Many hard-looking calculus questions become routine after one algebraic rewrite. Complete squares to identify conics, substitute u=x² for even quartics, and divide an improper rational function before partial fractions.

Preserve restrictions from the original expression. Cancelling a factor can simplify a formula without restoring an excluded point.

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Exam method

  • Compare polynomial degrees before decomposing a rational function.
  • Factor completely over the real numbers required by the question.
  • For x⁴+bx²+c, substitute u=x² and solve the quadratic in u.
  • For a conic, group x-terms and y-terms, complete both squares, then read the signs.
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Algebra before calculus: worked examples

2 questions

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Algebra before calculusSelf-review

Worked example 1

Find the real zeros and a lowest-degree real factorization of x4−5x2−36.x^4-5x^2-36.
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Formulas used in this topic

  • Df∘g={x∈Dg:g(x)∈Df}D_{f\circ g}=\{x\in D_g:g(x)\in D_f\}
  • (f−1)′(b)=1f′(a)when f(a)=b(f^{-1})'(b)=\frac{1}{f'(a)}\quad\text{when }f(a)=b
  • ax2+bx+c=a(x+b2a)2+c−b24aax^2+bx+c=a\left(x+\frac{b}{2a}\right)^2+c-\frac{b^2}{4a}
  • P(x)=D(x)Q(x)+R(x),deg⁡R<deg⁡DP(x)=D(x)Q(x)+R(x),\qquad \deg R<\deg D
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