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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)