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Level 387

Inverse Functions


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f(x) = 4x+6
f⁻¹(x) = (x-6)/4
f(x) = 4x-6
f⁻¹(x) = (x+6)/4
f(x) = -4x+6
f⁻¹(x) = -(x-6)/4
f(x) = -4x-6
f⁻¹(x) = -(x+6)/4
f(x) = 6x+4
f⁻¹(x) = (x-4)/6
f(x) = -¼x+6
f⁻¹(x) = -4(x-6)
f(x) = -¼x-6
f⁻¹(x) = -4(x+6)
one-to-one function
A function where each element of the range is paired with exactly one element of the domain.
Horizontal Line Test
inverse of a function is also a function if and only if no horizontal line intersects more than one
inverse function
Let f be a one-to-one function with domain A and range B. Then its ________________ f^-1 has domain B and range A and is defined by f^-1 (y) = x, f(x)=y for any y in B
Domain of f^-1
Range of f
Range of f^-1
Domain of f
Cancellation Equations
f^-1 [(f(x))] = x
Axis of Symmetry for Inverse Functions
The graph of f^-1 is obtained by reflecting the graph of f about the line y=x
Continuity of Inverse Functions
If f is a one-to-one continuous function defined on an interval, then its inverse function f^-1(x) is also continuous
(f^-1)'(a) = 1/f'(f^-1(a))
Derivative of an Inverse Function
Natural Logarithm Function
lnx = int. 1/t dt (1,x), x>0
(ln[u])'
u'/u
ln(xy)
lnx + lny
ln(x/y)
lnx - lny
ln(x^r)
rlnx
lim as x approaches ∞ of lnx
-∞
lim as x approaches 0+ of lnx
int. (1/x)dx
ln[x] + C
int. (tanx)dx
-ln[cosx] + C
Inverse of ln(x)=y
e^(y) = x
Natural Logarithmic Cancellation Equations
e^(lnx) = x, x>0
0
limit as x approaches -∞ of e^x
limit as x approaches ∞ of e^x
e^(x+y)
e^xe^y
e^(x-y)
e^x/e^(y)
(e^x)^r
e^rx
(e^u)'
(e^u)du
int. (e^u)du
e^u + C
a^x
e^xlna
a^(x+y)
a^(x)a^(y)
a^(x-y)
a^(x)/a^(y)
(a^x)^y
a^xy
(ab)^x
a^(x)b^(x)
(a^u)'
u'a^(u)lna
int. (a^u)du
(a^u)/lna + C
loga(x)=y
a^y = x
Loga (x)
lnx/lna
(loga(u))'
u'/ulna
e
limit as n approaches ∞ of (1+(1/n))^n
dy/dt = ky
Exponential Growth/ Decay Differential Equation
y(t) = y(0)e^kt
Exponential Growth/ Decay Equation
Newton's Law of Cooling Equation
T(t) = Ts + (T(0) -Ts)e^kt
dT/dt = k(T-Ts)
Newton's Law of Cooling Differential Equation
[f(x)^g(x)]'
Logarithmic Differentiation
sin^-1x = y
siny=x
Domain of arcsinx
Range of sinx, [-1,1]
Range of arcsinx
Domain of sinx, [-π/2, π/2]
Arcsinx Cancellation Equations
sin^-1(sinx)=x, [-π/2, π/2]
(sin^-1(u))'
du/sqrt.1-u^2
cos^-1x=y
cosy=x, [0,π]
Domain of arccosx
Range of cosx, [-1,1]
Range of arccosx
Domain of cosx, [0,π]
Arccosx Cancellation Equations
cos^-1(cosx)=x, [-1,1]
(cos^-1(u))'
-du/sqrt. 1-u^2
tan^-1x=y
tany=x, (-π/2, π/2)
Domain of arctanx
Range of tanx, (-∞,∞)
Range of arctanx
Domain of tanx, (-π/2, π/2)
π/2
limit as x approaches ∞ of tan^-1x
-π/2
limit as x approaches -∞ of tan^-1x
(tan^-1(u))'
du/1+u^2
(csc^-1(u))'
-du/usqrt.u^2 - 1
(sec^-1(u))'
du/usqrt.u^2 - 1
(cot^-1(u))'
-du/1+u^2
int. (du/sqrt. 1-u^2)
sin^-1(x) + C
int. (du/1+u^2)
tan^-1(u) + C
int. (du/a^2 + u^2)
1/a tan^-1 (u/a) + C
Only "one-to-one" functions have inverse
If x₁ ≠ x₂, then f(x₁) ≠ f(x₂) for all x
Horixontal Line Test
does it cross in more than one place
Special Inverses
f(x) =e∧x
e^x(e^y)
e^x+y
e^x-y
e^x / e^y
(e^x)^y
e^xy
ln(x^y)
y ln (x)