1. Barnett/Ziegler/Byleen Precalculus: A Graphing Approach
Chapter Four
Inverse Functions: Exponential and Logarithmic Functions

2. Operations on Functions
The sum, difference, product, and quotient of the functions f and g are the functions defined by
(f + g)(x) = f(x) + g(x) Sum function
(f – g)(x) = f(x) – g(x) Difference function
  (fg)(x) = f(x) g(x) Product function
  = Quotient function
Each function is defined on the intersection of the domains of f and g, with the exception that the values of x where g(x) = 0 must be excluded from the domain of the quotient function.
The composite of f and g is the function defined by
(f  g) (x) = f [g(x)] Composite function
The domain of f  g is the set of all real numbers x in the domain of g for which g(x) is in the domain of f.
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3. One-to-One Functions Horizontal Line Test
A function is one-to-one if no two ordered pairs in the function have the same second component and different first components.
Horizontal Line Test
A function is one-to-one if and only if each horizontal line intersects the graph of the function in at most one point.
(a) f(a) = f(b) for a  b; (b) Only one point has ordinate f is not one-to-one f(a); f is one-to-one
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4. Increasing and Decreasing
Functions
If a function f is increasing throughout its domain or decreasing throughout its domain, then f is a one-to-one function.
(a) An increasing function
is always one-to-one
(c) A one-to-one function is not
always increasing or decreasing
(b) A decreasing function
is always one-to-one
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5. Inverse of a Function
If f is a one-to-one function, then the inverse of f, denoted f –1, is the function formed by reversing all the ordered pairs in f. Thus,
f –1 = { (y, x) | (x, y) is in f }
To find the inverse of a function f:
Step 1. Find the domain of f and verify that f is one-to-one. If f is not one-to-one, then stop, since f –1 does not exist.
Step 2. Solve the equation y = f(x) for x. The result is an equation of the form x = f –1(y).
Step 3. Interchange x and y in the equation found in Step 2. This expresses f –1 as a function of x.
Step 4. Find the domain of f –1. Remember, the domain of f –1 must be the same as the range of f.
Check your work by verifying that
f –1 [ f(x) ] = x for all x in the domain of f , and
f [ f –1 (x)] = x for all x in the domain of f –1
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6. Basic Properties of the Graph of f(x) = bx, b > 0, b  1
Exponential Graphs
Basic Properties of the Graph of f(x) = bx, b > 0, b  1
1. All graphs will pass through the point (0, 1) since b0 = 1.
2. All graphs are continuous curves, with no holes or jumps.
3. The x axis is a horizontal asymptote.
4. If b > 1, then bx increases as x increases.
5. If 0 < b < 1, then bx decreases as x increases.
6. The function f is one-to-one.
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7. The Number e x è ç æ ø ÷ ö 1 + 2 10 …
100 …
1,000 …
10,000 …
100,000 …
1,000,000 … .
e = π e
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8. The Exponential Function with Base e
For x a real number, the equation f(x) = ex defines the exponential function with base e. The graphs of y = ex and y = e – x are shown in the figure.
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9. Exponential Growth and Decay
Description Equation Graph Uses
Short-term population
growth (people, bacteria, etc.);
growth of money at continuous
compound interest
Radioactive decay: light
absorption in water, glass, etc.;
atmospheric pressure;
electric circuits
Unlimited
growth
Exponential
decay
y = cekt c, k > 0
y = ce–kt c, k > 0
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10. Exponential Growth and Decay
Description Equation Graph Uses
y = c(1 – e–kt ) c, k > 0
Limited
growth
Logistic
Learning skills; sales fads;
company growth; electric circuits
Long-term population growth;
epidemics; sales of new products;
company growth y = M 1 + ce
–kt
c, k, M
> 0
4-4-38(b)

11. Logarithmic Function with Base 2y
= 2 x y y = x 10 f x y
= 2 –3 1 8 –2 4 –1 2 3
Ordered
pairs
reversed – f 1 y 5 x
= 2 or y
= log2x x –5 5 10 –5
DOMAIN of f
= (– ,  ) = RANGE of f –1
RANGE of f
= (0,  ) = DOMAIN of f –1
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12. Properties of Logarithmic Functions
If b, M, and N are positive
real numbers, b  1,
and p and x
are real numbers, then:
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13. The Decibel Scale
The decibel level D of a sound of intensity I , measured in watts per square meter (W/ m2) is given by
where I0 = 10–12 W/ m2 is the intensity of the least audible sound that an average healthy person can hear.
Sound Intensity, W/ m2 Sound
  1.0  10–12 Threshold of hearing
5.2  10–10 Whisper
3.2  10–6 Normal conversation
8.5  10–4 Heavy traffic
3.2  10–3 Jackhammer
1.0  100 Threshold of pain
8.3  102 Jet plane with afterburner
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14. The Richter Scale
The magnitude M on the Richter scale of an earthquake that releases energy E , measured in joules, is given by
where E0 = joules is the energy released by a small reference earthquake.
Magnitude on Richter scale Destructive power
M < 4.5 Small
4.5 < M < 5.5 Moderate
5.5 < M < 6.5 Large
6.5 < M < Major
7.5 < M Greatest
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15. Change-of-Base Formula
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