SECTION 5.2: Power, Rational, and Piecewise Functionswalkere/math140/notes/5.2.pdf · 2018. 7....
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MATH 140, Chapter 5 9 SECTION 5.2: Power, Rational, and Piecewise Functions Rational Functions Definition: Rational Functions A ___________________ function is the quotient (fraction) of two polynomial functions. The ___________________ of a rational function is all real numbers which do not make the denominator equal to 0. Example 1: Domain of a rational function Find the domain of the following rational functions. ) = 2 ! + − 1 − 3 ) = 3 ! + 2 ! + + 1 2 ! + 5 − 3 Example 2: Adding Rational Functions Rewrite each of the following as a single rational expression and simplify. ) 2 + 3 − 2 + 4 + 1
Transcript of SECTION 5.2: Power, Rational, and Piecewise Functionswalkere/math140/notes/5.2.pdf · 2018. 7....
MATH 140, Chapter 5 9
SECTION 5.2: Power, Rational, and Piecewise Functions Rational Functions Definition: Rational Functions
A ___________________ function is the quotient (fraction) of two polynomial functions.
The ___________________ of a rational function is all real numbers which do not make the
denominator equal to 0.
Example 1: Domain of a rational function Find the domain of the following rational functions.
𝑎) 𝑓 𝑥 =2𝑥! + 𝑥 − 1
𝑥 − 3
𝑏) 𝑓 𝑥 =3𝑥! + 2𝑥! + 𝑥 + 12𝑥! + 5𝑥 − 3
Example 2: Adding Rational Functions Rewrite each of the following as a single rational expression and simplify.
𝑎) 2𝑥 + 3𝑥 − 2 +
4𝑥 + 1
MATH 140, Chapter 5 10
𝑏) 1
2 𝑥 + ℎ − 3−1
2𝑥 − 3
Example 3: Evaluating Rational Functions Compute the following for the function defined below. Simplify your answers.
𝑓 𝑥 =2𝑥 − 13𝑥 + 2
a) 𝑓 −1 b) 𝑓(𝑥 + ℎ) c) 𝑓 𝑥 + ℎ − 𝑓(𝑥)
MATH 140, Chapter 5 11
Power Functions Definition: Power Functions A ______________________ function is a function 𝑓 of the form 𝑓 𝑥 = 𝑘𝑥!, where k and r are
real numbers.
Some examples include:
Review of Exponents:
• If 𝑎 ≠ 0, then: 𝑎! =_______, 𝑎!! =______, and in general, 𝑎!! =_______.
• 𝑎!! =_______________, 𝑎
!! =_________________, and in general, 𝑎
!! =__________.
• 𝑎!! =_______________=__________________.
Domain of nth roots:
The domain of 𝑓 𝑥 = 𝑔(𝑥)! is all real numbers (−∞,∞) if n is _____________.
The domain of 𝑓 𝑥 = 𝑔(𝑥)! is all numbers x such that 𝑔 𝑥 ≥ 0 if n is ______________.
Example 4: Domain of Power Function Find the domain of each of the following power functions: a) 𝑥! − 4! b) 𝑥 − 4 c) 𝑥! − 4
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𝑑) 2𝑥 − 3𝑥 − 4!
𝑒) 𝑥
2𝑥 + 4 Example 5: Rationalizing Functions Rationalize the denominator of !!!
!!!.
Example 6: Evaluating a Power Function For 𝑓 𝑥 = 3𝑥 − 1, a) Find 𝑓 2 b) Find 𝑓(𝑥 + ℎ)
MATH 140, Chapter 5 13
Piecewise Functions Definition: Piecewise Functions Functions whose definition involves more than one rule is called a ________________ function. To graph a piecewise function, graph each rule over the stated domain. Example 7: Absolute Value Function Graph the function 𝑓 𝑥 = |𝑥|. Recall, the absolute value of a number x, denoted |𝑥|, is the distance from x to 0. The absolute value function can be written as a piecewise function: 𝑓 𝑥 = 𝑥 = 𝑥, 𝑥 ≥ 0
−𝑥, 𝑥 < 0 Example 8: Writing Absolute Value as a Piecewise Function Express 𝑓 𝑥 = |2𝑥 + 5| as a piecewise defined function. Example 9: Graphing Piecewise Function Graph the following function:
𝑓 𝑥 =−𝑥, 𝑥 < 02, 0 ≤ 𝑥 < 2
𝑥 − 2, 𝑥 ≥ 2
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Example 10: Evaluating a Piecewise Function
For 𝑓 𝑥 = 𝑥 − 2, 𝑥 ≤ 2𝑥! + 4, 𝑥 > 2
a) Calculate 𝑓(−2) b) Calculate 𝑓(2) c) Calculate 𝑓(3) Example 11: Domain of a Piecewise Function
Find the domain of: 𝑓 𝑥 =!
!!!!, 0 ≤ 𝑥 < 3
4𝑥 − 1, 𝑥 ≥ 3
Example 12: Overtime In a given week, your job at Kyle Field pays $12/hour for the first 20 hours work and then $18/hour for any time worked over 20 hours. Express the money you earn A as a function of the hours x you work.
