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Slide 1 ___________________________________ 3.2 INVERSE FUNCTIONS AND LOGARITHMS
DEFINITION: ONE‐TO‐ONE FUNCTIONS
A function is a one‐to‐one function (or 1‐1 function) if for each value of y in the range there is only one corresponding value of x in the domain.
___________________________________ A function f one‐to‐one function if it never takes on the same value twice; that is, f(x1) ≠ f(x2) whenever x1 ≠ x2
___________________________________ A = {(2,3), (4,5), (8, 7)}
A is a one-to-one function.
___________________________________ B = {(3,1),(7,5),(6,4),(2,1)} B is not a one-to-one function
because even though 2 ≠ 3, f(2) = f(3) = 1.
___________________________________ ___________________________________ ___________________________________ Slide 2 ___________________________________ HORIZONTAL LINE TEST A function is one‐to‐one if and only if no horizontal line intersects its graph more than once.
f(x) = 3x
f(x) =
4
ex+e-x
2
___________________________________ 4
2
2
-5
5
-5
5
-2
-2
___________________________________ -4
-4
f(x) = x3
is one‐to‐one
e x + e− x
f (x) =
2
is not one-to-one
___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 3 ___________________________________ AN INVERSE FUNCTION IS A FUNCTION THAT UNDOES ANOTHER FUNCTION.
Only one‐to‐one functions have inverse functions.
___________________________________ If f is a one‐to‐one function with ordered pairs of the form (x, y), then its inverse function, denoted as f‐1, is also a one‐to‐one function with ordered pairs of the form (y, x).
___________________________________ Find the inverse of A = {(2, 4), (8, 7), (-1, 9)}
___________________________________ Ans {(4, 2), (7, 8), (9, -1)}
___________________________________ ___________________________________ ___________________________________ Slide 4 ___________________________________ HOW TO FIND THE INVERSE OF A ONE‐TO‐ONE FUNCTION f:
1. Write y in place of f(x).
___________________________________ 2. Interchange x and y.
3. In the new equation, solve for y in terms of x.
___________________________________ 4. Substitute f‐1(x) for y. (This new function is the inverse of f.)
___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 5 Example: Find the inverse of f(x) = 3-e2x
___________________________________ ___________________________________ Ans : f −1( x ) =
___________________________________ 1
ln(3 − x 2 )
2
___________________________________ ___________________________________ ___________________________________ Slide 6 Find the inverse of f(x) = e x
___________________________________ 3
___________________________________ Ans : y = 3 ln x
___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 7 ___________________________________ LAWS OF LOGARITHMS
If x and y are positive numbers, then
1. loga (xy) = loga x + logay
___________________________________ 2. loga ⎛⎜ x ⎞⎟ = loga x − loga y
⎝y⎠
3. loga (xr ) = rloga x (where r is any real number) ___________________________________ CHANGE OF BASE FORMULA For any positive number a (a ≠ 1), we have
loga x =
___________________________________ ln x
ln a
___________________________________ ___________________________________ ___________________________________ Slide 8 ___________________________________ Example 1: Solve the equation for x.
e5‐3x = 10 (hint: take ln of both sides)
___________________________________ ___________________________________ x ≈ 0.8991
Example 2 Evaluate log8 5
log8 5 =
___________________________________ ln5
≈ 0.773976
ln 8
___________________________________ ___________________________________ ___________________________________ Slide 9 ___________________________________ Example 3: Express the quantity as a single logarithm.
lnx + alny ‐ blnz
Ans
___________________________________ ⎛ xy a ⎞
ln ⎜ b ⎟
⎝ z ⎠
___________________________________ Example 4: Solve the equation ln(5 – 2x) = ‐3
Ans
x=
e-3 − 5
5 - e-3
or x =
−2
2
___________________________________ ___________________________________ ___________________________________ ___________________________________