Download 5.2 - Rational, Power, and Piecewise-Defined Functions

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Kathryn
Bollinger, March 15, 2017
5.2 - Rational, Power, and Piecewise-Defined Functions
Def: A rational function has the form
f (x) =
n(x)
d(x)
where n(x) and d(x) are polynomials and d(x) 6= 0. The domain of a rational function is the set of
all real numbers such that d(x) 6= 0.
Ex: Find the domain of the following functions using interval notation.
(a) f (x) =
x2 − x − 2
x2 − 9
(b) g(x) =
2x
x−4
(c) h(x) =
x2 + 6x + 8
x2 − x − 6
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Kathryn
Bollinger, March 15, 2017
A rational expression is a sum, difference, product or quotient of terms, each which is the quotient
of polynomials. To add or subtract rational functions or expressions, you must first get a common
denominator.
Ex: Compute and simplify the following.
(a)
x−8
x+3
−2
x−5
x−5
(b)
2x + 9 x + 1
+
x+3
x−2
The difference quotient is the quantity
in calculus.
f (x + h) − f (x)
and is a very important concept used
h
Ex: Find and simplify the difference quotient for the following functions.
(a) f (x) = −x2 + 2x − 5
(b) g(x) =
4
3x + 7
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Kathryn
Bollinger, March 15, 2017
Power Functions
A power function is of the form f (x) = kxr , where k and r are any real numbers.
Some specific cases...
• Case 1: r = n, where n is a positive integer
◦ A polynomial with only one term.
◦ Domain:
• Case 2: r =
1
, where n is a positive integer
n
◦ A function of the form f (x) = x1/n =
√
√
◦ f (x) = xm/n = n xm = ( n x)m
√
n
x, known as a radical (or root) function.
◦ Parent Function Examples:
y = x1/2 =
√
y = x1/3 =
x
◦ Domain:
Domain:
• Case 3: r = −1
◦ A function of the form f (x) = x−1 =
1
, known as the reciprocal function.
x
◦ Its graph is a hyperbola.
◦ Domain:
The power r can be any real number...there are many other possibilities.
√
3
x
4
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Kathryn
Bollinger, March 15, 2017
Ex: Write the following in its equivalent radical function format.
(a) x2/5 =
(b) x7/2 =
(c) (2x + 1)3/11 =
When finding the domain of a function involving radical functions, remember that you cannot take
an even root of a negative number.
If an even root is in the denominator, then what you are taking the root of cannot be negative and
it also cannot be zero.
If your function involves an odd root, there are no domain issues unless the odd root is in the
denominator. When this is the case, what you are taking the odd root of cannot be zero.
Ex: Find the domain, using interval notation, of each of the following functions.
(a) f (x) =
√
(b) g(x) =
√
5
3x + 8
2x − 7
(c) h(x) = √
6
2x
5x + 9
(d) f (x) = √
7
√
x
x−3
5
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Kathryn
Bollinger, March 15, 2017
√
4
x+8
(e) g(x) =
(2x + 3)(x − 6)
Rationalizing the numerator or denominator of a fraction means to get rid of any radicals in that
part of the fraction. This often involves multiplying by the conjugate.
Ex: Rationalize the denominator and simplify the function below.
x−9
f (x) = √
x+3
Ex: Find and simplify the difference quotient for the function g(x) =
√
4x + 9.
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Kathryn
Bollinger, March 15, 2017
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Piecewise-Defined Functions: Functions whose domains are divided into two or more parts
with a different rule applied to each part. To graph, graph each rule over the appropriate portion
of the domain.


 x+6
, x < −5
Ex: For g(x) =
9
, −5 ≤ x < 6

 −2x + 4 , x ≥ 6
find the following:
(a) g(−8)
(b) g(−5)
(c) g(0)
(d) g(6)
(e) g(8)
Ex: For f (x) =
(
0.5x2
, −4 ≤ x < −1
−4 + x , x ≥ −1
(a) find f (−3), f (−1), f (0), and f (−10).
(b) make an accurate graph of f (x).
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Kathryn
Bollinger, March 15, 2017
Ex: Sketch an accurate graph of the following function: g(x) =


 x+1
, x ≤ −2
4
, −2 < x < 1

 −3x − 2 , x ≥ 1
When finding the domain of a piecewise-defined function, first check the domain intervals and see
if any values of x do not have a rule identified. Then, check each function piece to make sure it is
defined everywhere on that interval.
Ex: Find the domain, using interval notation, for each of the following functions.
(a) h(x) =

x+3




 x+2





(b) g(x) =
,x≤0
x+7
(x − 9)(x + 4)
 √
5

x






 5x

x+3






 √3x + 6
,x>0
,x<0
,2<x<4
,x≥6
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Kathryn
Bollinger, March 15, 2017
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An important example of a piecewise-defined function is the absolute value function f (x) = |x|
which is defined by
(
−x , x < 0
f (x) =
x
,x≥0
Ex: Write g(x) = |x − 2| as an equivalent piecewise-defined function.
Ex: Write h(x) = |3x + 7| as an equivalent piecewise-defined function.
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Kathryn
Bollinger, March 15, 2017
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Ex: A taxi cab company in a certain town charges all customers a base fee of $5.00 per ride.
They then charge an additional 50 cents/mile for mileage over 5 miles up to 10 miles traveled and
$1/mile for each mile over 10 miles. Write a piecewise function, C(x), for the cost of a cab ride if
x represents the number of miles traveled.