Section 1.6 Inverse Functions and Logarithms
An Inverse Function "undoes" the original function!
If N=f(t) gives the number of bacteria in a culture after t hours, then
t=f­1(N) gives the number of hours to produce a specific number of bacteria.
A function must be "one­to­one" in order to have an inverse that is also a function. A function is "one­to­one" if it never takes on the same value twice. (For every "x" there is only one, unique "y" and for every "y" there is only one, unique "x."
Given: The Inverse Function:
• f is a one­to­one function f­1(y)=x f(x) =y
• Domain is A Domain is B
• Range is B Range is A
These Compositions Verify that functions are inverses and are
sometimes called "Inverse Properties" or "cancellation equations."
• f-1(f(x)) = x for every x in A
• f(f-1(x)) = x for every x in B
Steps for finding an Inverse Function: (textbook differs slightly...)
1) replace f(x) with y
2) interchange x and y in the equation
3) solve this equation for y
4) replace y with f-1(x) notation.
Example: Find the inverse of f(x) = x3 + 2
Graphs of Inverses:
• Interchange x and y coordinates
• Symmetrical with respect to the line y=x (reflected about this line)
Example: Sketch the graph of f(x) =√(x­2) and its inverse:
Logarithmic Function: the inverse of an exponential function!
logax = y ay = x
Properties Review:
Inverse 1:
Inverse 2:
Product:
Quotient:
Power:
Natural Log: logex = lnx lnx = y ey = x
(revisit inverse properties with ln)
Graphs of log functions and ln: (Inverse)
Recall for Exponential Functions: Now Logarithmic Functions:
Domain (­∞, +∞) Domain (0,∞)
Range (0, ∞) Range (­∞, +∞)
Y­intercept (0, 1) x­intercept (1,0)
Horizontal Asymptote y=0 (x­axis) Vertical Asymptote x=0 (y­axis)
Increasing when a>1 (same)
Decreasing when 0<a<1 (same)
Constant when a=1 (same)
One­to­one Function (same)
Brief Review:
Solve: lnx = 5
Solve: e
Express lna + 1/2(lnb) as a single logarithm:
Change of Base Formula: logax = lnx
lna
Evaluate log8 5 correct to six decimal places:
5-3x
= 10
What is the inverse of the following function?
f(x) = sin x