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In summary: A function is a special type of relation where each input has only one output. The domain is the set of inputs and the range is the set of outputs. Quadratic functions are a specific type of function with a parabolic graph. The domain of combined functions is found by considering the restrictions of individual functions.
5. Quadratic Function: A quadratic function is a function of the form:
f(x) = ax² + bx + c, where 'a', 'b', and 'c' are constants and a ≠ 0.
* Example: f(x) = 2x² - 3x + 1 is a quadratic function.
* Vertex Form: A quadratic function can also be written in vertex form: f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola.
* Domain of a Quadratic Function: The domain of a quadratic function is always all real numbers (-∞, ∞). The range depends on whether 'a' is positive (opens upwards) or negative (opens downwards).
4. Domain of Combination Functions:
When combining functions (e.g., addition, subtraction, multiplication, division, composition), the domain of the resulting function is the intersection of the domains of the individual functions, with additional restrictions for division and composition.
* Example: If f(x) = 1/x (domain: x ≠ 0) and g(x) = x - 1 (domain: all real numbers), then the domain of (f + g)(x) = 1/x + x - 1 is x ≠ 0. The domain of (f/g)(x) = 1/(x(x-1)) is x ≠ 0 and x ≠ 1.
3. Domain and Range:
* Domain: The set of all possible x-values (inputs) of a relation or function. * Range: The set of all possible y-values (outputs) of a relation or function.
* Example: For the function f(x) = x + 2, the domain is all real numbers (-∞, ∞), and the range is all real numbers (-∞, ∞).
2. Function: A function is a special type of relation where *each* x-value is associated with *only one* y-value.
* Example: {(1, 2), (3, 4), (5, 6)} is a function. {(1,2), (1,3)} is NOT a function.
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