Editing Image (mathematics) (section)

==Examples==
# <math>f : \{ 1, 2, 3 \} \to \{ a, b, c, d \}</math> defined by <math>
    \left\{\begin{matrix}
      1 \mapsto a, \\
      2 \mapsto a, \\
      3 \mapsto c.
    \end{matrix}\right.
  </math>{{paragraph break}} The ''image'' of the set <math>\{ 2, 3 \}</math> under <math>f</math> is <math>f(\{ 2, 3 \}) = \{ a, c \}.</math> The ''image'' of the function <math>f</math> is <math>\{ a, c \}.</math> The ''preimage'' of <math>a</math> is <math>f^{-1}(\{ a \}) = \{ 1, 2 \}.</math> The ''preimage'' of <math>\{ a, b \}</math> is also <math>f^{-1}(\{ a, b \}) = \{ 1, 2 \}.</math> The ''preimage'' of <math>\{ b, d \}</math> under <math>f</math> is the [[empty set]] <math>\{ \ \} = \emptyset.</math>
# <math>f : \R \to \R</math> defined by <math>f(x) = x^2.</math>{{paragraph break}} The ''image'' of <math>\{ -2, 3 \}</math> under <math>f</math> is <math>f(\{ -2, 3 \}) = \{ 4, 9 \},</math> and the ''image'' of <math>f</math> is <math>\R^+</math> (the set of all positive real numbers and zero). The ''preimage'' of <math>\{ 4, 9 \}</math> under <math>f</math> is <math>f^{-1}(\{ 4, 9 \}) = \{ -3, -2, 2, 3 \}.</math> The ''preimage'' of set <math>N = \{ n \in \R : n < 0 \}</math> under <math>f</math> is the empty set, because the negative numbers do not have square roots in the set of reals.
# <math>f : \R^2 \to \R</math> defined by <math>f(x, y) = x^2 + y^2.</math>{{paragraph break}} The [[Fiber (mathematics)|''fibers'']] <math>f^{-1}(\{ a \})</math> are [[concentric circles]] about the [[Origin (mathematics)|origin]], the origin itself, and the [[empty set]] (respectively), depending on whether <math>a > 0, \ a = 0, \text{ or } \ a < 0</math> (respectively). (If <math>a \ge 0,</math> then the ''fiber'' <math>f^{-1}(\{ a \})</math> is the set of all <math>(x, y) \in \R^2</math> satisfying the equation <math>x^2 + y^2 = a,</math> that is, the origin-centered circle with radius <math>\sqrt{a}.</math>)
# If <math>M</math> is a [[manifold]] and <math>\pi : TM \to M</math> is the canonical [[Projection (mathematics)|projection]] from the [[tangent bundle]] <math>TM</math> to <math>M,</math> then the ''fibers'' of <math>\pi</math> are the [[tangent spaces]] <math>T_x(M) \text{ for } x \in M.</math> This is also an example of a [[fiber bundle]].
# A [[quotient group]] is a homomorphic ''image''.