MATH 131-503 Fall 2015
1.6
c
Wen
Liu
1.6 Inverse Functions and Logarithms
A function f is called a one-to-one function if it never takes on the same value twice; that is,
f (x1 ) 6= f (x2 )
whenever x1 6= x2
Horizontal Line Test: A function is one-to-one if and only if no horizontal line intersects its graph
more than once.
Example 1: (p. 62) Determine if f (x) = x3 and f (x) = x2 are one-to-one.
Let f be a one-to-one function with domain A and range B. Then its inverse function f −1 has
domain B and range A and is defined by
f −1 (y) = x ⇐⇒ f (x) = y
for any y in B.
1
1
. We write
= (f (x))−1 instead.
f (x)
f (x)
The letter x is traditionally used as the independent variable, so when we concentrate on f −1 rather
than on f , we usually reverse the roles of x and y in the definition and write
Be careful: f −1 (x) 6=
f −1 (x) = y ⇐⇒ f (y) = x
By substituting for y in the definition and substituting for x above, we get the following cancellation
equations:
f −1 (f (x)) = x for every x ∈ A
f f −1 (x) = x for every x ∈ B
Example 2: If f (1) = −5, f (3) = 7, and f is one-to-one, find f −1 (−5) and f −1 (f (3)).
Page 1 of 6
MATH 131-503 Fall 2015
1.6
c
Wen
Liu
How to Find the Inverse Function of a One-to-One Funtion f :
1 Write y = f (x)
2 Solve this equation for x in terms of y (if possible)
3 To express f −1 as a function of x, interchange x and y. The resulting equation is y = f −1 (x).
Examples:
3. Given f (x) = x2 + 9.
(a) Is the function one-to-one? If not, restrict the domain so that the function is one-to-one.
(b) State the domain and range for the function.
(c) Find f −1 (x).
(d) State the domain and range for f −1 (x).
Page 2 of 6
MATH 131-503 Fall 2015
4. Find the inverse of f (x) =
1.6
c
Wen
Liu
6x − 1
.
2x + 5
Note: The graph of f −1 is obtained by reflecting the graph of f about the line y = x.
Example 6: (p. 64) Sketch the graphs of f (x) =
coordinate axes.
√
−1 − x and its inverse function using the same
Page 3 of 6
MATH 131-503 Fall 2015
1.6
c
Wen
Liu
Logarithmic Functions
If a > 0 and a 6= 1, the exponential function f (x) = ax is either increasing or decreasing and so it is
one-to-one by the Horizontal Line Test. It therefore has an inverse function f −1 , which is called the
logarithmic function with base a and is denoted by loga . If we use the formulation of an inverse
function given above, we have
loga x = y ⇐⇒ ay = x
Note:
loga (ax ) = x for every x ∈ R
aloga x = x for every x > 0
Laws of Logarithms: If x and y are positive numbers, then
1. loga (xy) = loga x + loga y
x
2. loga
= loga x − loga y
y
3. loga (xr ) = r loga x, where r is any real number
Example 7: Find the exact value of log2 6 − log2 15 + log2 20.
Natural Logarithms
The logarithm with base e is called the natural logarithm and has a special notation:
loge x = ln x
Then we have
ln x = y ⇐⇒ ey = x
ln(ex ) = x x ∈ R
eln x = x x > 0
ln e = 1
Page 4 of 6
MATH 131-503 Fall 2015
1.6
c
Wen
Liu
Examples:
8. Express the given quantity as a single logarithm. ln(a + b)2 + ln(a − b) − 5 ln c.
9. Solve each equation for x.
(a) e9x+a − 9 = 0 for some constant a.
(b) ln(2 − 6x) + m = 0 for some constant m.
Page 5 of 6
MATH 131-503 Fall 2015
10. Find (a) the domain of f (x) =
c
Wen
Liu
1.6
√
3 − e2x and (b)f −1 and its domain.
Change of Base Formula: For any positive number a (a 6= 1), we have
loga x =
ln x
ln a
Example 11: (p. 87) Evaluate log8 5 correct to six decimal places.
Page 6 of 6