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Calculus I (MAT 145) Dr. Day Thur sday Jan 17,2013

Calculus I (MAT 145) Dr. Day Thur sday Jan 17,2013. Function Characteristics (continued) (1.1 – 1.3) Assignments. Functions (PIP p5-8; Text section 1.1-1.3). Define One-to-one functions Domain and Range Representations for Functions Increasing and Decreasing Functions; Notation

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Calculus I (MAT 145) Dr. Day Thur sday Jan 17,2013

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  1. Calculus I (MAT 145)Dr. Day Thursday Jan 17,2013 • Function Characteristics (continued) (1.1 – 1.3) • Assignments MAT 145

  2. Functions(PIP p5-8; Text section 1.1-1.3) • Define • One-to-one functions • Domain and Range • Representations for Functions • Increasing and Decreasing Functions; Notation • Families of Functions • Operating on Functions MAT 145

  3. Functions A function f is a rule that assigns to each element x in a set A exactly one element, called f(x), in a set B. MAT 145

  4. One-to-One Functions Afunction f is a rule that assigns to each element x in a set A exactly one element, called f(x), in a set B. A one-to-one function is a function for which each element y in set B corresponds to exactly one element x in set A. MAT 145

  5. Domain and Range of Functions Afunction f is a rule that assigns to each element x in a set A exactly one element, called f(x), in a set B. • Set A is the domain of the function. • Set B is the range of the function. • What does f(x) represent? Limitations on a Domain and Range MAT 145

  6. Representations of Functions Graphsprovide a visual representation of the relationship between input (x) and output (y)by plotting ordered pairs (x,y), such that , y = f(x),on orthogonal (right-angled) axes. • Sketch the graphs and label at least three ordered pairs for these functions. MAT 145

  7. Representations of Functions Represent each of the functions above: • in a table • verbally State the domain and range for each function. MAT 145

  8. Increasing and Decreasing Functions MAT 145

  9. Function Notation • What does f(x) = 3x + 5 represent? • Using y = f(x) as just described, determine and simplify each of the following. • f(2) (b) f(−4) (c) f() (d) f(x-1) • f(2x+4) (f) f(t2+7t) (g) f(f(x)) (h) f(x + h) (i) f(x + h) – f(x) MAT 145

  10. Families of Functions Algebraic—functions formed by using algebraic operations (+, -, x, ÷, nth root of a , a to the mth power , such that m and n are rational) on coefficients and variables • Polynomial (Constant, Linear, Quadratic, Cubic, Quartic, and others) • Power • Root • Rational Transcendental—non-algebraic functions (i.e., those that “transcend” algebra) • Trigonometric • Exponential • Logarithmic • Others—Piecewise, Absolute Value, Greatest Integer, and others MAT 145

  11. Operating on Functions Algebra of functions: apply one or more of the four basic operations (+,−,,) to two or more functions. Composition of functions: putting together two ormore functions in a particular way. When two functions are composed, one is evaluated within the other. MAT 145

  12. Reminders/Assignments • Assignments • PIP Packet #54 • WebAssign: Sec 1.2 and 1.3 • Quiz #3: function characteristics (1.1 to 1.3) MAT 145

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