Download Chapter Eight: Rational Functions and Radical Functions Section

Chapter Eight: Rational Functions and Radical Functions
Section One: Inverse, Joint, and Combined Variation
In this lesson we will discuss three other ways that variables can relate to each other. We discussed
direct variations in chapter one ( y  kx ).
In an inverse variation the two variables are related by the equation xy  k or y 
k
where k is again
x
called the constant of variation. If we have a single point and know that an equation varies inversely, we
can easily find the constant of variation.
EX1: The variable y varies inversely as x, and y is 22.5 when x is 6.2. Find the constant of variation, and
write the equation for the relationship. Then find y when x is 0.1, 0.2, and 0.3.
The graph of an inverse variation is not continuous but has two asymptote lines.
A joint variation is a type of direct variation. In a joint variation, one variable varies directly as two
variables, y  kxz , where k is again the constant of variation.
EX2: y varies jointly as x and z. Write the appropriate joint-variation equation, and find y for the given
values of x and z.
a. y  16 when x  4 and z  0.5 ; x  2 and z  0.25
When more than one type of variation occurs in the same equation, we call it a combined variation.
EX3: Write an equation for the following combined variations
a. z varies jointly as x and y and inversely as r
b. x varies directly as the square of y and inversely as the cube root of z
EX4: The number of hours it takes men to assemble machines varies directly as the number of machines
and inversely as the number of men. If four men can assemble 12 machines in four hours, how many
men are needed to assemble 36 machines in eight hours?
Section Two: Rational Functions and Their Graphs
EX1: William begins with 75 mL of a 15% acid solution. He adds x mL of distilled water to the container
holding the acid solution. Write a function that represents the acid concentration of the solution in
terms of x. What is the acid concentration of the solution if 35 mL of distilled water is added?
The function in the previous example is a rational function (ratio=fraction). A rational function is a
function that is the quotient of two polynomials. When working with rational functions it is very
important to keep in mind that division by zero is undefined. Therefore, we cannot have zeros on the
bottom of a fraction. Values that are not in the domain of a function are referred to as excluded values.
3x 2  x  2
EX2: Find the domain of f ( x)  2
x  2x  3
Rational functions are not continuous. When dealing with rational functions we encounter lots of
asymptotes. An asymptote was a limit line for a function. It gets closer and closer to the line but never
touches it.
Vertical asymptotes sometimes occur at excluded values of a rational function. If x  a is a factor of the
bottom of the function but not a factor of the top, then we have a vertical asymptote at x  a .
EX3: Identify the vertical asymptotes of the function g ( x) 
x3
. Check with a graph.
x  x6
2
Horizontal asymptotes can be referred to as end behavior. It is the line that the graph approaches as x
goes on to positive and negative infinity. We check the degree of the numerator and denominator and
use the following three rules to determine if we have a horizontal asymptote:
1. num  den : y  0 is the horizontal asymptote
a
is the horizontal asymptote where a and b are leading coefficients of the fraction
b
2.
num  den : y 
3.
num  den : there is no horizontal asymptote
EX4: Find all the horizontal and vertical asymptotes of the function h( x) 
2x2  2x 1
. Check with a
x 2  x  12
graph.
Asymptotes sometime prove very helpful when trying to sketch complicated graphs.
2x 1
. Then divide the function and
x4
1
describe the transformation from the parent function f ( x )  .
x
EX5: Use the asymptotes to help sketch the graph of r ( x) 
A rational function can also have a hole in it. A hole is simply a jump in the graph over a single point. A
hole occurs when x  a is a factor of both the top and the bottom of your fraction.
EX6: Identify all asymptotes and holes in the following graphs.
a.
b.
3x 2  x3
x2  2 x  3
3x  28  x 2
f ( x)  2
x  12 x  35
f ( x) 
Section Three: Multiplying and Dividing Rational Expressions
The number one rule that we have to remember for this lesson: factors can be canceled and terms
cannot. If multiple terms are being added, you cannot cancel out just a single term from the expression.
x 2  7 x  18
EX1: Simplify the following fraction by factoring:
x 2  81
When multiplying fractions, remember that we can cross cancel
EX2: Multiply the following rational expressions
9 x5 6 5 x
 
5 x 7 18
x 2  4 x  12 x 2  2 x  35
b.

x 2  11x  30
x4
a.
We divide fractions by flipping the second fraction and multiplying
EX3: Simplify
x 2  25
x5
 2
2
x  2 x  3 x  3x  18
A complex fraction is simply a division problem written as a fraction. It looks like a fraction with fractions
in the numerator and denominator. It has one major fraction bar and two minor fraction bars.
x2  4
2
EX4: Simplify the complex fraction x  2 x  1
x2
x 1
EX5: Simplify the following expressions
1  7 x 1  18 x 2
1  4 x 2
( x  y) y 1  2 x( x  y)1
b.
( x  y) y 1  2 x( x  y)1
a.
Section Four: Adding and Subtracting Rational Expressions
Before two fractions can be added or subtracted, the denominators must be the same. We then simply
add the numerators and leave the denominators. Be sure to check and see if the fraction can be
reduced!
EX1: Simplify. Identify any excluded values.
a.
b.
9x
2x

x7 x7
7x
4 x  12
(make sure to distribute the minus sign)

x4
x4
Ask yourself, “How can I make the bottoms the same.” That value is your Least Common Denominator
(LCD).
EX2: Simplify. Identify any excluded values.
a.
b.
x
50
 2
x  5 x  25
x
x
 2
3x  3 x  1
EX3: Write each complex fraction as a rational expression. Then simplify. Identify any excluded values.
a.
a
a
b.
1
a
2
3
5
a
a

a
1
a
2

3
5
a
EX4: Angela drove from home to work at an average speed of 65 mph. When she left work at 5:00 P.M.,
she drove home at an average speed of 50 mph. What was her average speed over the entire trip?
EX5: Simplify
a.
b.
1  xy 1
x 1  y 1
1 1

a 2 b2
a 2  2(ab) 1  b 2
Partial Fractions?
Section Five: Solving Rational Equations and Inequalities
A rational equation can be solved using the following set of steps:
1. Find the LCD for the entire equation
2. Multiply every term by the LCD. All fractions should cancel.
3. Clean up the mess.
4. Solve the remaining equation.
5. Check to make sure the solution(s) is not an excluded value. Another name for solutions that
don’t check out is extraneous solutions.
Note: Check first to see if you have a simple proportion. If yes, cross products are equal.
EX1: Solve the following equations
a.
b.
c.
x
6

x  3 x 1
x
8
2
 2

x2 x 4 x2
x
x
3

 2
x  2 x  3 x  5x  6
We solve rational inequalities by the same steps we have used for all inequality problems.
1. Find and plot critical values
a. Solve the corresponding equation to find these
b. Excluded values are also critical values
2. Pick a number from each region on the number line
3. Test these numbers by plugging them into the original inequality
4. Graph accordingly
EX2: Solve the rational inequalities
a.
b.
c.
x 1
3
x2
2
3

x 1 x  4
a 3
1
1
 2

3a
3a  9a a  3
Section Six: Radical Expressions and Radical Functions
Remember that from previous lessons the symbol x stands for the principle root of a number. We
sometimes call it a radical. We can only find the square root of positive x’s if we are only dealing in the
real domain. The square root of a negative is an imaginary number. The graph of f ( x) 
below.
x is shown
EX1: Find the domain of the function. Check with a graph.
a.
r ( x)  4 x  7
b. t ( x) 
x2  4x  3
The same rules for transformations from section 2.7 apply to radical functions as well.
EX2: Describe the transformations from the parent function f ( x) 
a.
f ( x)  3 x  2  5
b.
f ( x)   2 x  1  8
x.
Also from previous lessons, the inverse of a function is found by switching the x and the y or the domain
and the range. When working with radical functions we sometimes have to use a trick with the quadratic

b  b 2  4ac 
formula  x 
 to get y by itself.


2a


EX3: Find the inverse of the following functions. Check with a graph.
a. f ( x)  x 2  1
b.
f ( x)  x 2  3x  4
EX4: You can use the formula D( x)  1.5 x to approximate the distance, D, in miles that you can see
from a height, x, in feet. Find the maximum distance you can see from heights of 10 feet, 20 feet, and 30
feet.
The cube root of a number,
example,
3
3
x , is a number that when multiplied by itself three times gives you x. For
64 is 4 since 4  4  4  64 Therefore, the same way that f ( x)  x is the inverse of
f ( x)  x 2 , we can say that f ( x)  3 x is the inverse of f ( x)  x3 . We can find the cube root of
8  2 ) and positive numbers (i.e. 3 8  2 ). Therefore, the domain of the
cube root function f ( x)  3 x includes all real numbers (the range is also all real numbers). Below we
negative numbers (i.e.
3
see the graph of f ( x) 
3
x.
EX5: Evaluate each expression
a.
6 3 8  2
b.
4

3
216

2
A radical expression is an expression with at least one radical symbol. As we have seen, the radical
symbol could be a square root or a cube root. However, we could go even farther than that and include
fourth roots, fifth roots, sixth roots, etc. The process for finding these roots are the same. These were
covered somewhat in section 2.2. In that section we learned that a fractional exponent was another way
to write a radical. We have to use this method in order to type higher order radicals into most
calculators.
1
1
16  16 2
EX6: Evaluate the following radicals
a.
5
1024
b.
4
93
3
27  27 3
1
1
4
81  814
10
236  23610
Section Seven: Simplifying Radical Expressions
Before working our first example we need to review the properties of radicals from section 5.2.
1.
ab  a  b
2.
a
a

b
b
EX1: Use the radical properties to simplify each expression
81x 3 y 4 z 6
a.
125x 6 yz 5
b.
3
c.
 64ab5  2  8ab6
1
d.
9 3 48 x8
1
 2 x3  3
We can add or subtract radicals only if they are similar. For example, 3 2  5 2  8 2 but
3 2  4 3 cannot be added
EX2: Simplify each sum or difference
a.
b.
3 
4 
 

27    15  48 
24  8  96
When multiplying radicals remember that
a  b  ab
EX3: Simplify each product
a.
b.
 2  3 3 9  7 3 
8 2  8  3 2  7 32 
As mentioned in previous section, math etiquette says that we should remove radicals from the bottom
of fractions (this includes i ’s). We do so by multiplying by the radical if it is a single term or multiplying
by its conjugate if it is a binomial.
EX4: Rationalize the denominator
2
6
4
b.
3 7
a.
Section Eight: Solving Radical Equations and Inequalities
The important rule to remember when solving radical equations is “isolate or separate.” If we have a
single radical symbol, isolate it by moving everything else to the other side. If we have two radicals,
separate them. Move them to opposite sides of the equal sign. We then use powers to remove the
radical(s). Always check your answers. Extraneous answers occur a lot when dealing with radical
equations.
Note: The main thing to remember is that when you raise a binomial to a power, you must FOIL!
EX1: Solve the following equations
a.
4 2 x  1  12
b. 3 3 x2  5x  14  12  6
c.
4 x  2  3x  1
d.
x  3x  4  2
We solve radical inequalities the same way we solve all inequalities.
1. Find and plot critical values
a. Solve the corresponding equation to find these
b. Since we can only find square roots of positives, a critical value occurs where the stuff
under the radical is equal to zero since it is the division between positive and negative.
We can however take an odd root of negatives.
2. Pick a number from each region on the number line
3. Test these numbers by plugging them into the original inequality
4. Graph accordingly
EX2: Solve the inequalities
x7  4
b. 4 x  1  x (use a calculator to find the roots)
a.
3
c.
2 x 2  2  3x  7  0