Download Section 3.1

Functions
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Overview
• Review what is a function
• Domain and Range
• Piecewise functions
• Maxima, Minima, Turning Points
• Increasing and Decreasing functions
• Average rate of change of functions
• Graphing:
– Translation
– Symmetry
– Scale
• Even and Odd Functions
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A Function
• A rule that goes from x to y, or x to f(x)
• Must be unique – for each value of x, have only one f(x)
• Vertical Line Test: a vertical line can intersect the graph of a
function at only one point
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Domain and Range
• Domain
Where a function gets its inputs, the x value
• Range
The outputs of a function
• Specifying a domain may
– Exclude points for which a function is not defined
– Restrict the input values such that we have a function
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Functions to Watch For
• Rational Functions:
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5−𝑥
• Square Roots: 1 − 5𝑥
• Logarithms: log (1-x)
• Each of these functions has restricted domains
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Examples
Find the domain of
y = 3𝑥 + 9
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y= 2
𝑡 −6𝑡 −7
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Solution
Find the domain of
y = 3𝑥 + 9
If we are looking for y to be a real number, we need to
have 3x + 9 ≥ 0. Our domain is x ≥ -3
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y= 2
𝑡 −6𝑡 −7
We need to avoid dividing by zero, so our domain is
all real numbers, excluding 7 and -1
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Example
Find the domain of
𝑥+3
𝑥 −4
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𝑥+3
𝑥 −4
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Solution
• For both of these expressions, we need x ≠4
• For the first, we also want the value under the square root
to be ≥ 0. This restricts x to be x ≤ -3 υ x > 4
𝑥+3
𝑥 −4
• For the second, there is no restriction that the value be ≥ 0, so
we have only x ≠4
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𝑥+3
𝑥 −4
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Finding the Range
Find the range of
y=
𝑥+2
𝑥 −3
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Solution
Solve for x:
𝑥+2
y=
𝑥 −3
y(x-3) = x + 2
xy – x = 3y + 2
3𝑦+2
x=
𝑦 −1
We can see that for any y there is a x, excluding y = 1. However,
if y = 1 is a solution, that means 2 = -3, which is impossible
The range is all reals excluding y = 1
NOTE: This technique does not always work!
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Possible Errors
• f(a+b) ≠ f(a)+f(b)
• f(ab) ≠ f(a)f(b)
• f(1/a) ≠ 1/f(a)
• f(a) / f(b) ≠ a/b or f(a/b)
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Average Rate of Change
• The average rate of change of a function on an interval [a, b]
is the slope of the line connecting (a, f(a)) and (b, f(b))
• m=
𝑓 𝑎 −𝑓(𝑏)
𝑎 −𝑏
=
∆𝑦
∆𝑥
, the change in y divided by the change in x
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Calculating the Average Rate of Change
Find the average rate of change,
𝑓 𝑎 −𝑓(𝑏)
𝑎 −𝑏
for the interval given
f(x) = 4x2 on the interval [x, 3]
f(x) = x2 on the interval [1, 1+h]
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Solution
Find the average rate of change,
𝑓 𝑎 −𝑓(𝑏)
𝑎 −𝑏
for the interval given
f(x) = 4x2 on the interval [x, 3]
4(9 − 𝑥 2 )
3 −𝑥
= 4(3 + x)
f(x) = x2 on the interval [1, 1+h]
(1+ℎ)2 −1
ℎ
=
1+2ℎ+ℎ2 −1
ℎ
=2+h
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Arithmetic Operations
• (f + g)(x) = f(x) + g(x)
• (f - g)(x) = f(x) - g(x)
• (fg)(x) = f(x)g(x)
• (f / g)(x) = f(x) / g(x), for g(x) ≠ 0
• The domain is all inputs belonging to both f and g, with the
added provision that in the division g(x) ≠ 0
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Composition of Functions
If f(x) = 3x and g(x) = x - 1
• f(g(x)) = 3(x-1)
We also write this as f  g(x)
The domain of f  g(x) is those inputs for g for which g(x) is in
the domain of f
Note: In general, f  g(x) ≠ g  f(x)
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Example
f(x) = x2 + 3 and g(x) = 𝑥
What is the domain of f(g(x))?
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Solution
• f(g(x)) = x2 +3, but the domain is not all real numbers.
• The domain of g is all positive reals, hence that is the domain
of f (g(x))
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More on Domain and Composition
𝑓 𝑥 =
1
𝑥+2
and 𝑔 𝑥 =
𝑥
𝑥−3
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One to One
• A function is one-to-one if each element in the range of comes
from a unique element of the domain of f
For example, f(x) = x2 is not one-to-one, since f(-1) = f(1)
• Algebraically: show f(a) = f(b) iff a = b
• Geometrically – the horizontal line test
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Inverse Functions
• If a function, f is one-to-one, then the inverse of f(x),
f -1(x) is a function such that f -1(f(x)) = x for x in the domain
of f, and f(f -1 (y))= y for y in the domain of f -1
• Note: the range of the inverse is equal to the domain of the
original function; the domain of the inverse is the range of the
original
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Example
• f(x) = 𝑥 . Find f -1 (y), its domain and range
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Solution
• x = y2, so f -1 (y) = y2
• What is the domain of f -1? Need y  0 for x to be one-to-one
• Remember, the range of the inverse is equal to the domain of
the original function; the domain of the inverse is the range of
the original
• The domain of the original function, f(x) = 𝑥, is x  0, so this
is the range of the inverse, f -1 (y)
• The range of f(x) is x  0, and this is the domain of the inverse
– we require for f -1 (x) that x  0
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Find the inverse of f(x)
• f(x) =
2𝑥+3
𝑥 −1
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Solution
• f(x) =
2𝑥+3
𝑥 −1
• y = (2x + 3)/(x – 1)
• yx – y = 2x + 3
• x(y – 2) = 3 + y
• x = (3 + y)/(y-2)
• Range and domain: We can’t have y = 2 in the domain of the
inverse. Is it in the range of f?
2 = (2x+3)/(x-1), 2x -2 = 2x + 3. But -2 is never 3!
• Can 1 be in the range of the inverse?
1 = (3+x)/(x-2), x – 2 = 3 + x, again, it doesn’t work
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