MATH 151 Engineering Math I, Spring 2014
JD Kim
Week9 Section 4.2, 4.3
Section 4.2 Inverse Functions
Definition
One-to-One
A function f with domain A is called a One-to-One function if no two elements
of A have the same image; that is
f (x1 ) 6= f (x2 ) whenever x1 6= x2
or
f (x1 ) = f (x2 ) then x1 = x2 .
Do the horizontal test
If a horizontal line intersects the graph of f in more than one point; there are
number x1 and x2 such that f (x1 ) = f (x2 ), f is not one-to-one.
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Ex1) Is the function f (x) = x2 − 2x + 5 is one-to-one?
Ex2) Is the function g(x) = 5 − 4x3 is one-to-one?
2
Ex3) Is the function f (x) =
x−2
is one-to-one?
x+2
Ex4) How can we restrict the domain of f (x) = cos x to make it one-to-one?
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Definition Inverse Fundtion
Let f be a one-to-one function with domain A and range B. Then the inverse
exists, 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.
domain of f −1 = range of f
range of f −1 = domain of f
Cancellation equations
f −1 (f (x)) = x for every x in A
f (f −1 (x)) = x for every x in B
How to find the inverse function of a one-to-one function f .
Step1 Write y = f (x).
Step2 Solve this equation for x in terms of y (if possible)
Step3 Interchange x and y. The resulting equation if y = f −1 (x).
The graph of f −1 is obtained by relecting the graph
of f about the line y = x.
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Ex5) Find the inverse function and the domain and range of the inverse function.
5-1) y = 5 − 4x3
5-2) f (x) =
2x + 1
1 − 3x
5
5-3) f (x) = x2 + x, for x ≤ −
1
2
Theorem
If f is a one-to-one differentiable function with inverse function g = f −1 and
f ′ (g(a)) 6= 0, then the inverse function is differentiable at a and
g ′ (a) =
1
f ′ (g(a))
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Ex6) Suppose g is the inverse of f and f (2) = 3, f ′ (2) = 7, f (3) = 4 and
1
f ′ (3) = . Find g ′(3).
2
Ex7) Suppose g is the inverse of f . Find g ′(4) if f (x) = 3 + x + ex .
Ex8) Suppose g is the inverse of f . Find g ′(2) if f (x) =
7
√
x3 + x2 + x + 1.
Section 4.3 Logarithmic Functions
Definition Logarithmic function
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. 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
f −1 (x) = y ⇔ f (y) = x,
then we have
loga x = y ⇔ ay = x.
Thus, if x > 0, loga x is the exponent to which the base a must be raised to give x.
Ex9) Evaluate the following;
9-1) log2 8
9-2) log3 81
9-3) log3
1
81
8
9-4) log16 4
Cancellation
The cancellation equation, when applied to f (x) = ax and f −1 (x) = loga x,
become
loga (ax ) = x for every x ∈ R,
aloga x = x for everyx > 0.
Ex10) What is the domain and range of f (x) = log2 x.
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Theorem If a > 1, the function f (x) = loga x is a one-to-one, continuous,
increasing function with domain (0, ∞) and range R. If x, y > 0, then
1. loga (xy) = loga x + loga y
x
2. loga ( ) = loga x − loga y
y
3. loga (xy ) = y loga x
4. loga a = 1
5. aloga x = x
6. change of base:
1
logc b
or loga b =
loga b =
logc a
logb a
Definition Natural Logarithm, Common Logarithm
We define the Natural Logarithm to be loge x denoted by ln x. We define the
Common Logarithm to be log10 x, denoted by log x.
Ex11) Evaluate the following
11-1) log
1
10
1
11-2) ln √
e
10
Ex12) Solve each equation for x.
12-1) log2 x = 3
12-2) 2 log(x + 1) = 3
12-3) ex = 16
12-4) e3−x + 8 = 14
12-5) 2log2 3 + ln(x − 2) = 13
Ex13) Find the domain and range of f (x) = ln(4 − x2 ).
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Ex14) Find each limit;
14-1) limx→5+ ln(x − 5)
14-2) limx→∞ ln(x2 − x)
14-3) lim
θ→(
π log(cos θ)
)−
2
14-4) limθ→0+ log(cos θ)
Ex15) Express the given quantity as a single logarithm.
15-1) log2 x + 5 log2 (x + 1) +
1
log2 (x − 1)
2
15-2) ln x + a ln y − b ln z
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Ex16) Solve the following equations for x.
16-1) ln x − ln(x − 1) = 1
16-2) log2 x + log2 (x + 2) = 3
Ex17) Find the limite;
17-1) limx→∞ (ln(3x2 − 2x + 5) − ln(2x2 + 4x))
17-2) limx→∞ (ln(3x2 ) − ln(6x4 − 3x + 1))
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Ex18) Find the inverse function of each function.
18-1) f (x) = ln(x + 2)
18-2) g(x) =
10x
10x + 1
Ex19) Find an equation of the tangent to the curve y = e−x that is perpendicular
to the line 2x − y = 4.
Ex20) (Change of Base) Evaluate log8 5 correct to six decimal places.
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