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1.2 Real Vocab
| Term | Definition |
|---|---|
| Function | Let A and B be any two sets. A function f from A into B is a subset of A x B with the property that each x ∈ A is the first component of precisely one ordered pair (x, y) ∈ f. |
| Domain | The set A is called the domain of f, denoted dom f. |
| Codomain | B is called the codomain of f. |
| Range | The range of f, denoted Range f, is defined by Range f = {y ∈ B: (x, y) ∈ f for some x ∈ A}. |
| Onto | If Range f = B, then the function f is said to be onto B. |
| Real-valued function on A | If f: A -> ℝ, then f is said to be a real-valued function on A |
| Projection from A x B to A | Let A and B be two nonempty sets and consider the projection function p from A x B to A defined by p = {((a, b), a): (a, b) ∈ A x B} |
| Identity function on A | Let A be any nonempty set and let i = {(x, x): x ∈ A}, i is a function from A onto A whose value at each x ∈ A is x; that is, i(x) = x. |
| Image | Let f be a function from A to B. If E ⊆ A, the image of E under f is defined by f(E) = {y ∈ B: x ∈ E and f(x) = y} (image ⊆ B) |
| Pre-image | If H ⊆ B, the pre-image of H, denoted f-1(H) is defined by f-1(H) = {x ∈ A: f(x) ∈ H} If H = {y} we write f-1(y), thus for y ∈ B f-1(y) = {x ∈ A: f(x) = y} (pre-image ⊆ A) |
| One-to-one | A function f from A into B is said to be one-to-one if whenever x1 ≠ x2, the f(x1) ≠ f(x2) |
| Inverse function | If f is a one-to-one function from A onto B, let f-1 = {(y, x) ∈ B x A: f(x) = y} The function f-1 from B onto A is called the inverse function of f. Furthermore, for each y ∈ B, x = f-1(y) iff f(x) = y. |
| Composition of functions | If f is a function from A to B and g is a function from B to C, then the function g∘f: A -> C is defined by g∘f = {(x, z) ∈ A x C: z = g(f(x))} is called the composition of g with f |
| f maps A to B | f: A -> B indicates that f is a function from A into B, or that f maps A to B or is a mapping of A to B |
Created by:
Mollygsmithh
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