Free VCE Mathematical Methods Flashcard Review
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Define a function
A relation where each element in the domain maps to exactly one element in the codomain. Every input has precisely one output.
Define range vs codomain
Codomain is the set of all possible outputs specified for the function. Range is the set of actual outputs produced. Range is a subset of codomain.
Define one-to-one (injective) function
A function where distinct inputs map to distinct outputs. If f(a) = f(b), then a = b. Graphically passes the horizontal line test.
Define onto (surjective) function
A function where every element in the codomain is mapped to by at least one element in the domain. The range equals the codomain.
Define inverse function f-1
The inverse of f is f-1 such that f-1(f(x)) = x and f(f-1(y)) = y. A function has an inverse if and only if it is bijective.
How to find an inverse function equation
Replace f(x) with y, swap x and y, then solve for y. The graph of f-1 is the reflection of f across the line y = x.
Define composite function (f ∘ g)(x)
(f ∘ g)(x) = f(g(x)). The output of g becomes the input of f. Domain requires x in domain of g and g(x) in domain of f.
General form of polynomial function
f(x) = anxn + an-1xn-1 + ... + a1x + a0, where n is non-negative integer degree and an ≠ 0. Degree determines end behaviour.
End behaviour of polynomials
Even degree, positive leading coefficient: both ends up. Even degree, negative: both ends down. Odd degree, positive: left down, right up. Odd degree, negative: left up, right down.
General form of exponential function
f(x) = abˣ + c, where a ≠ 0, b > 0, b ≠ 1. Horizontal asymptote is y = c. If b > 1 and a > 0, function increases.
General form of logarithmic function
f(x) = a logᵦ(x) + c, where a ≠ 0, b > 0, b ≠ 1. Vertical asymptote is x = 0. Domain: (0, ∞). Logarithmic is inverse of exponential.
Three key logarithm properties
Product: logᵦ(mn) = logᵦ(m) + logᵦ(n). Quotient: logᵦ(m/n) = logᵦ(m) - logᵦ(n). Power: logᵦ(mn) = n logᵦ(m).
Change of base formula for logarithms
logᵦ(x) = ln(x) / ln(b) = log(x) / log(b), where ln is natural logarithm and log is base-10. Allows conversion between logarithm bases.
Define power function
f(x) = axn, where a ≠ 0 and n is a real constant. If n > 0 and not an integer, domain is x ≥ 0. If n < 0, domain is x ≠ 0.
Properties of sine function
f(x) = sin(x): period 2π, amplitude 1, domain: all reals, range [-1, 1]. Zeros at x = nπ for integer n. No vertical asymptotes.
Properties of cosine function
f(x) = cos(x): period 2π, amplitude 1, domain: all reals, range [-1, 1]. Zeros at x = π/2 + nπ for integer n. No vertical asymptotes.
Properties of tangent function
f(x) = tan(x): period π, vertical asymptotes at x = π/2 + nπ (integer n), domain: all reals except asymptotes, range: all reals.
Describe function transformations f(x) → af(k(x - h)) + v
a: vertical dilation by |a|. k: horizontal dilation by 1/|k|. h: horizontal translation right by h. v: vertical translation up by v. Negative a or k causes reflections.
Definition of the derivative f'(x)
f'(x) = lim[h→0] (f(x+h) − f(x))/h. Represents instantaneous rate of change of f at x, or slope of tangent line to the curve.
Power rule for differentiation
If f(x) = xn, then f'(x) = nxn-1 for any real n. Generalisation: if f(x) = axn, then f'(x) = anxn-1.