Lecture Notes Basic Functions and their Properties page 1

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1 Lecture Notes Basic Functions and their Properties page De nition: A function f is (or injective) if for all a and b in its domain, if a = b, then f (a) = f (b). Alternative de nition: A function f is (or injective) if for all a and b in its domain, if f (a) = f (b), then a = b. De nition: A function f is increasing on an interval I if for all a and b in I, if a < b, then f (a) f (b). De nition: A function f is strictl increasing on an interval I if for all a and b in I, if a < b, then f (a) < f (b). De nition: A function f is decreasing on an interval I if for all a and b in I, if a < b, then f (a) f (b). De nition: A function f is strictl decreasing on an interval I if for all a and b in I, if a < b, then f (a) > f (b). De nition: A function f is even if for all in its domain, f ( is smmetrical to the ais. ) = f (). The graph of an even function De nition: A function f is odd if for all in its domain, f ( ) = f (). The graph of an odd function is smmetrical to the origin. Note that while an integer is either even or odd, most functions are neither even, nor odd. functions are sort of rare but these properties are ver useful for us. Even and odd De nition: A rational function is a quotient of two polnomial functions. The concept of a continuous function is ver important. Although this term will not be precisel de ned, the intuitive idea of a continuous function is thatwe can draw its graph without lifting the pencil. For eample, f () = is a continuous function but g () = is not; it is not continuous at =. c Hidegkuti, Powell, Last revised: October,

2 Lecture Notes Basic Functions and their Properties page Basic Functions.) Linear functions f () = m + b where m =. The graph is a straight line. One useful form of the equation can be obtained b factoring out the slope: f () = m + b = m + b m m > m < Case. If m > Case. If m < domain: R domain: R range: R range: R intercept: (; b) intercept: (; b) intercept: b m ; no maimum or minimum strictl increasing intercept: b m ; no maimum or minimum strictl decreasing c Hidegkuti, Powell, Last revised: October,

3 Lecture Notes Basic Functions and their Properties page.) Quadratic functions f () = a + b + c where a =. The graph is a parabola. It opens upward if a > and opens downward if a <. a > a < Case. If a > Case. If a < Eample: f () = + Eample: f () = + standard form: f () = ( + ) standard form: f () = ( ) + factored form: f () = ( + ) ( ) factored form: f () = ( ) ( ) domain: R range: [ ; ) domain: R range: ( ; ] intercept: (; ) intercept: ; intercepts: ( ; ) and (; ) intercepts: (; ) and (; ) not not no maimum no minimum minimum: ( ; ) maimum: (; ) strictl decreasing on ( ; ) strictl increasing on ( ; ) strictl increasing on ( ; ) strictl decreasing on (; ) c Hidegkuti, Powell, Last revised: October,

4 Lecture Notes Basic Functions and their Properties page.) Monomials f () = n n is even n is odd Case. If n is even Case. If n is odd domain: R range: [; ) domain: R range: R intercept: (; ) intercept: (; ) intercept: (; ) intercept: (; ) not no maimum no minimum or maimum minimum: (; ) strictl increasing on R strictl decreasing on ( ; ) strictl increasing on (; ) black graph: f () = black graph: f () = red graph: f () = red graph: f () = c Hidegkuti, Powell, Last revised: October,

5 Lecture Notes Basic Functions and their Properties page.) The rational functions f () = and g () = f () = g () = domain: R n fg range: R n fg domain: R n fg range: (; ) no intercept no intercept no intercept no intercept not no maimum or minimum no minimum or maimum strictl decreasing on ( ; ) and on (; ) strictl increasing on ( ; ) and strictl decreasing on (; ) not continuous at = not continuous at = vertical asmptote: the line = vertical asmptote: the line = horizontal asmptote: the line = horizontal asmptote: the line = c Hidegkuti, Powell, Last revised: October,

6 Lecture Notes Basic Functions and their Properties page.) Radical functions f () = np n is even n is odd Case. If n is even Case. If n is odd domain: [; ) range: [; ) domain: R range: R intercept: (; ) intercept: (; ) intercept: (; ) intercept: (; ) no maimum no minimum or maimum minimum: (; ) strictl increasing strictl increasing continuous on (; ) black graph: f () = p black graph: f () = p red graph: f () = p red graph: f () = p c Hidegkuti, Powell, Last revised: October,

7 Lecture Notes Basic Functions and their Properties page.) Eponential functions f () = a where a > a > < a < Case. If a > Case. If < a < domain: R range: (; ) domain: R range: (; ) no intercepts no intercepts intercept: (; ) intercept: (; ) no maimum or minimum no maimum or minimum strictl increasing strictl decreasing horizontal asmptote: = horizontal asmptote: = c Hidegkuti, Powell, Last revised: October,

8 Lecture Notes Basic Functions and their Properties page.) Logarithmic functions f () = log a where a > and a = a > < a < Case. If a > Case. If < a < domain: (; ) range: R domain: (; ) range: R no intercepts no intercepts intercept: (; ) intercept: (; ) no maimum or minimum no maimum or minimum strictl increasing strictl decreasing vertical asmptote: = vertical asmptote: = continuous on (; ) continuous on (; ).) Absolute value function, f () = jj domain: R range: [; ) intercept: (; ) ; intercept: (; ) not no maimum minimum: (; ) strictl decreasing on (; ) and strictl increasing on (; ) For more documents like this, visit our page at and click on Lecture Notes. questions or comments to mhidegkuti@ccc.edu. c Hidegkuti, Powell, Last revised: October,

Math RE - Calculus I Functions Page 1 of 10. Topics of Functions used in Calculus

Math RE - Calculus I Functions Page 1 of 10. Topics of Functions used in Calculus Math 0-03-RE - Calculus I Functions Page of 0 Definition of a function f() : Topics of Functions used in Calculus A function = f() is a relation between variables and such that for ever value onl one value.