1.4_Functions.pdf

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10/22/23 1 SFT3113 Mathematics for Physics SHAHRUL KADRI BIN AYOP Teaser Sketch a graph for the function ζ ࠵? = 2 sgn ࠵? ! . 1 SFT3113 Mathematics for Physics SHAHRUL KADRI BIN AYOP Functions At the end of this lesson, you will be able to translate a function along the ࠵? -axis. recognize even and odd functions. apply Heaviside and signum functions. 2
10/22/23 2 SFT3113 Mathematics for Physics SHAHRUL KADRI BIN AYOP Functions ࠵? = 2࠵?࠵? ࠵? = 2࠵? ࠵? ࠵? = ࠵? ࠵? Processor ࠵? Input ࠵? Output ࠵? independent variable dependent variable coefficient ࠵? is a function of ࠵? Example: ࠵? = ࠵?࠵? + " ! ࠵?࠵? ! ࠵? = ࠵? ࠵? 3 SFT3113 Mathematics for Physics SHAHRUL KADRI BIN AYOP Translation of function along ࠵? -axis Example : ࠵? = 2 ࠵? translation by 1 unit For any positive number ࠵?, represent ࠵? ࠵? translation along ࠵? -axis to the left and right, respectively ࠵? = ࠵? ࠵? argument ࠵? ࠵? ࠵? ࠵? = 2 ࠵? + 1 ࠵? = 2 ࠵? − 1 ࠵? = 2࠵? + 2 ࠵? = 2࠵? − 2 ࠵? = ࠵? ࠵? + ࠵? ࠵? = ࠵? ࠵? − ࠵? 2 −2 4
10/22/23 3 SFT3113 Mathematics for Physics SHAHRUL KADRI BIN AYOP Even and odd functions ࠵? ࠵? even if ࠵? −࠵? = ࠵? ࠵? odd if ࠵? −࠵? = −࠵? ࠵? ࠵? ࠵? ࠵? = cos ࠵? ࠵? −࠵? = cos −࠵? ࠵? ࠵? = sin ࠵? ࠵? −࠵? = sin −࠵? 0 1 1 0 0 0.25࠵? 0.707 0.707 0.707 −0.707 0.50࠵? 0 0 1 −1 0.75࠵? −0.707 −0.707 0.707 −0.707 ࠵? −1 −1 0 0 even functions are symmetry about ࠵? -axis some functions are neither even nor odd ࠵? ࠵? ࠵? −࠵? −0.5࠵? 0.5࠵? ࠵? −180° −90° 90° 180° Examples: 5 SFT3113 Mathematics for Physics SHAHRUL KADRI BIN AYOP Heaviside function ࠵? ࠵? some functions are not smooth Heaviside function is a unit step function with a jump or discontinuity ࠵? ࠵? ࠵? ࠵? 0 when ࠵? < 0 1 when ࠵? ≥ 0 ࠵? ࠵? = inclusive! ࠵? ࠵? −࠵? # ࠵? ࠵? ࠵? # 0 when ࠵? < ࠵? ! 1 when ࠵? ≥ ࠵? ! ࠵? ࠵? −࠵? ! = Examples: 6
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