Algebraic fractions and functions flashcards

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Algebraic (rational) fraction
A polynomial divided by a polynomial, defined wherever the denominator is non-zero.
Improper algebraic fraction
One whose numerator has degree ≥ the degree of the denominator; divide to get a polynomial plus a proper fraction.
Function
A mapping in which every member of the domain is mapped to exactly one output.
Domain and range
Domain: the set of allowed inputs. Range: the set of outputs, written in terms of f(x), e.g. f(x) ≥ 3.
One-one function
Different inputs always give different outputs; a horizontal line meets the graph at most once.
Many-one function
At least two inputs share the same output, e.g. x ↦ x² with f(−2) = f(2). Still a function, but it has no inverse.
One-many mapping
One input gives more than one output (e.g. x ↦ ±√x). It is not a function.
Composite function fg
fg(x) = f(g(x)): apply g first, then f.
Inverse function f⁻¹
The function that reverses f: if f(a) = b then f⁻¹(b) = a. Exists only when f is one-one.
Self-inverse function
A function with f⁻¹ = f, i.e. ff(x) = x for every x in the domain.
Division identity for algebraic fractions
N(x)/D(x) ≡ Q(x) + R(x)/D(x), with degree of R less than degree of D. (No — learn it)
Dividing two fractions
(a/b) ÷ (c/d) = (a/b) × (d/c). (No — learn it)
Composite and inverse identities
fg(x) = f(g(x)); f⁻¹f(x) = ff⁻¹(x) = x. (No — learn it)
Domain and range of f⁻¹
Domain of f⁻¹ = range of f; range of f⁻¹ = domain of f. (No — learn it)
Completed square of ax² + bx + c
a(x + b/2a)² + c − b²/4a; the vertex gives the range of a quadratic and where to restrict its domain. (No — derive it)
Graph of y = f⁻¹(x) compared with y = f(x)
Reflection in the line y = x: (a, b) ↦ (b, a), and asymptotes x = k and y = k swap.
Graph of y = (3x + 5)/(x + 1) = 3 + 2/(x + 1)
Asymptotes x = −1 and y = 3; crosses the axes at (−5/3, 0) and (0, 5).
Graph after cancelling a common factor, e.g. (2x² + 5x − 3)/(4x² − 1)
Same curve as (x + 3)/(2x + 1) but with a hole at x = ½, where the cancelled factor (2x − 1) is zero.
Reading a range from a graph
Sketch the graph over the domain only and read off the y-values covered; check whether end values and asymptote values are included.
Restricted quadratic y = (x − 3)² − 4, x ≥ 3, and its inverse
Right half of the parabola from (3, −4); inverse y = 3 + √(x + 4) starts at (−4, 3), the reflection of the vertex.
Why you can only cancel factors
A factor multiplies the whole numerator and denominator; a term does not. (x + 6)/(x + 2) ≠ 3: at x = 1 it is 7/3.
Why only one-one functions have inverses
If two inputs shared an output, the reversed mapping would send that output to two values — not a function.
When can fg be formed?
When every output of g is an allowed input of f: the range of g must lie inside the domain of f.
Choosing the sign of a square root in an inverse
The outputs of f⁻¹ must lie in the (restricted) domain of f: x ≥ 3 needs +√, x ≤ 3 needs −√.
Why divide an improper fraction before finding a range
The form constant + proper fraction shows the asymptote, which is usually the excluded end of the range.
Simplify (2x² + 5x − 3)/(4x² − 1).
(x + 3)(2x − 1)/((2x − 1)(2x + 1)) = (x + 3)/(2x + 1).
Simplify 2/(x − 3) − (x + 7)/(x² − x − 6).
[2(x + 2) − (x + 7)]/((x − 3)(x + 2)) = (x − 3)/((x − 3)(x + 2)) = 1/(x + 2).
Divide: (2x⁴ − x³ + 7x² − x + 2)/(x² + 2).
2x² − x + 3 + (x − 4)/(x² + 2).
f(x) = 3x − 1, g(x) = x² + 2. Find fg(x) and gf(x).
fg(x) = 3x² + 5; gf(x) = (3x − 1)² + 2 = 9x² − 6x + 3.
Find f⁻¹ for f(x) = (2x + 3)/(x − 1), x > 1.
f(x) = 2 + 5/(x − 1) so range f(x) > 2; f⁻¹(x) = (x + 3)/(x − 2), x > 2.
Range of f(x) = x² − 4x + 7, x ∈ ℝ?
(x − 2)² + 3, so f(x) ≥ 3.
f(x) = x² − 6x + 5, x ≥ 3. Find f⁻¹(x).
(x − 3)² − 4 ⇒ f⁻¹(x) = 3 + √(x + 4), x ≥ −4.
Order in fg(3)
g first: fg(3) = f(g(3)). The function next to the input acts first.
f⁻¹(x) vs 1/f(x)
Different things: for f(x) = 2x, f⁻¹(x) = x/2 but 1/f(x) = 1/(2x).
Answering 'find f⁻¹(x)'
Give the rule AND its domain (the range of f). Without the domain the answer is incomplete.
Solutions of an equation involving a function
Check each root against the domain and reject any outside it, with a reason.
Subtracting a fraction with a two-term numerator
Bracket it: 2(x + 2) − (x + 7) = x − 3, not 2x + 4 − x + 7.

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