1. Inverse Functions
and Logarithms

2. One-To-One Functions
A function is one-to-one if two domain values correspond to the same range value.
Algebraically, a function is one-to-one if f (x1) ≠ f (x2) for all x1 ≠ x2.

3. One-To-One Functions
Graphically, a function is one-to-one if its graph passes the horizontal line test. That is, if any horizontal line drawn through the graph of a function crosses more than once, it is one-to-one.

4. Try This Determine if the following functions are one-to-one.
(a) f (x) = 1 + 3x – 2x 4
g(x) = cos x + 3x 2
(c)
(d)

5. f −1(x) = y <=> f (y) = x
Inverse Functions
The inverse of a one-to-one function is obtained by exchanging the domain and range of the function. The inverse of a one-to-one function f is denoted with f -1.
Domain of f = Range of f -1
Range of f = Domain of f -1
f −1(x) = y <=> f (y) = x

6. Try This
Sketch a graph of f (x) = 2x and sketch a graph of its inverse. What is the domain and range of the inverse of f.
Domain: (0, ∞)
Range: (-∞, ∞)

7. Inverse Functions
You can obtain the graph of the inverse of a one-to-one function by reflecting the graph of the original function through the line y = x.

8. Inverse of a One-To-One Function
To obtain the formula for the inverse of a function, do the following:
Let f (x) = y.
Exchange y and x.
Solve for y.
Let y = f −1(x).
Domain: (0, ∞)
Range: (-∞, ∞)

9. Inverse Functions
Determine the formula for the inverse of the following one-to-one functions.
(a)
(b)
(c)

10. Logarithmic Functions
The inverse of an exponential function is called a logarithmic function.
Definition: x = a y if and only if y = log a x

11. Logarithmic Functions
Domain: (0, ∞)
Range: (-∞, ∞)
Asymptote: x = 0
Increasing for a > 1
Decreasing for 0 < a < 1
Common Point: (1, 0)
The function f (x) = log a x is called a logarithmic function.

12. Logarithmic Functions
Now determine the inverse of g(x) = 3x.
Definition: x = a y if and only if y = log a x

13. Properties of Logarithms
log a (ax) = x
for all x  
a log a x = x
for all x > 0
log a (xy) =
log a x + log a y
log a (x / y) =
log a x – log a y
log a xn = nlog a x
Common Logarithm:
log 10 x = log x
Natural Logarithm:
log e x = ln x
All the above properties hold.

14. Properties of Logarithms
The natural and common logarithms can be found on your calculator. Logarithms of other bases are not. You need the change of base formula.
where b is any other appropriate base.

15. Try These Determine the exact value of log 8 2.
Determine the exact value of ln e 2.3.
Evaluate log to four decimal places.
Write as a single logarithm: ln x + 2ln y – 3ln z.
Solve 2x + 5 = 3 for x.

16. Try This
In the theory of relativity, the mass of a particle with velocity v is
where m0 is the mass of the particle and c is the speed of light in a vacuum. Find the inverse function of f and explain its meaning.