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Hyperbolic & Inverse Hyperbolic Functions
DEFINITIONS
Hyperbolic Functions Inverse Hyperbolic Functions
sinh(x) = (ex
– e-x
)/2 •sinh-1
x = ln (x + √(x2
+ 1))
cosh(x) = (ex
+ e-x
)/2 •cosh-1
x = ln (x + √(x2
– 1))
tanh(x) = (ex
– e-x
)/(ex
+ e-x
)= sinhx/coshx •tanh-1
x = ln [(1 + x)/(1 – x)]
coth(x) = (ex
+ e-x
)/(ex
– e-x
)= cosh x/sin hx =
1/tanh(x) •coth-1
x = ln [(x + 1)/(x – 1)]
sech(x) = 2/(ex
+ e-x
) = 1/cosh x •sech-1
x = ln [{1 + √(1 – x2
)}/x]
csch(x) = 2/(ex
– e-x
) = 1/sinh x •csch-1
x = ln [{1 + √(1 + x2
)}/x]](https://image.slidesharecdn.com/realfunction-graphdomainrange-250112181343-33d254e9/85/Real-Function-Graph-Domain-Range-pptx-22-320.jpg)
Real Function - Graph, Domain, Range.pptx
- 2. 2 Real Function NOTATION & MEANING We write the set is called the Domain of , the set is called the Codomain of , the set of all y-values taken by the function is called the Range of We write where ,
- 3. 3 Real Function CLASSIFICATION • Constant Function • Polynomial Function • Algebraic Function • Exponential Function • Logarithmic Function • Trigonometric Function • Inverse Trigonometric Function • Hyperbolic Function • Inverse Hyperbolic Function • Combinations of above types
- 4. Constant Function where is a constant, a fixed real number EXAMPLES Function Domain Codomain Range is called Zero Function. It is a special case of Constant Function
- 6. 6 Polynomial Functions, , where CLASSIFICATION value Function Name Domain Range Zero Function Linear Function Quadratic Function Cubic Function Polynomial Function of degree …. P(x) = an xn + an-1 xn-1 +.……….… + a0 , where a0, a1, ………, an are all constants General Polynomial Function of deg …. P(x) = an xn + an-1 xn-1 +.……….… + a0 , where the powers of x are all odd Odd Polynomial Function P(x) = an xn + an-1 xn-1 +.……….… + a0 , where the powers of x are all even Even Polynomial Function
- 7. 7 Polynomial Functions, , where GRAPHS
- 8. 8 CLASSIFICATION Odd Polynomial Function Even Polynomial Function Neither Even nor Odd ax ax2 + b ax + b General Linear Function ax3 + bx ax4 + bx2 + c ax2 + bx + c General Quadratic Function ax5 + bx3 + cx ax6 + bx4 + bx2 + c ax3 + bx2 + cx + d General Cubic Function Polynomial Functions (continued)
- 9. 9 Algebraic Functions, where EXAMPLES Value of a Function Domain Codomain Range a > 0 a > 0 a < 0 a < 0 a < 0
- 11. 11 Exponential Functions, where EXAMPLES Value of a Function Domain Codomain Range a > 0 a > 0 a > 0 or a = e a = e
- 13. 13 Logarithmic Functions, where EXAMPLES Value of a Function Domain Codomain Range a = 2 a = 0.56 a = 10 a = e Special Cases of Logarithmic Functions When , is called the Common Logarithm When , or is called the Natural Logarithm
- 15. 15 Trigonometric Functions, Function Domain Codomain Range
- 18. 18 Inverse Trigonometric Functions, Function Range Codomain Domain
- 20. 20 Hyperbolic Functions, Function Domain Codomain Range
- 21. 21 Inverse Hyperbolic Functions, Function Domain Codomain Range
- 22. 22 Hyperbolic & Inverse Hyperbolic Functions DEFINITIONS Hyperbolic Functions Inverse Hyperbolic Functions sinh(x) = (ex – e-x )/2 •sinh-1 x = ln (x + √(x2 + 1)) cosh(x) = (ex + e-x )/2 •cosh-1 x = ln (x + √(x2 – 1)) tanh(x) = (ex – e-x )/(ex + e-x )= sinhx/coshx •tanh-1 x = ln [(1 + x)/(1 – x)] coth(x) = (ex + e-x )/(ex – e-x )= cosh x/sin hx = 1/tanh(x) •coth-1 x = ln [(x + 1)/(x – 1)] sech(x) = 2/(ex + e-x ) = 1/cosh x •sech-1 x = ln [{1 + √(1 – x2 )}/x] csch(x) = 2/(ex – e-x ) = 1/sinh x •csch-1 x = ln [{1 + √(1 + x2 )}/x]
- 23. 23