Notes with answers

7.1 

Your Notes 

nth Roots and Rational 

Exponents 

Goal p Evaluate nth roots of real numbers. 

VocabulaRy 

nth root of a For an integer n greater than 1, if 

bn 5 a, then b is the nth root of a (Written as n 

Ï } 

a .) 

Index of a radical The integer n, greater than 1, in 

the expression n 

a 

Ï } 

NumbER of REal nth Roots 

Let n be an integer, (n > 1) and let a be a real number. 

If n is an even integer: If n is an odd integer: 

• a < 0 No real nth roots. • a < 0 One real nth root: 

n 

Ï } 

a 5 a 1/n 

• a 5 0 One real nth root: • a 5 0 One real nth root: 

n 

0 5 0 

n 

0 5 0 

Ï } 

Ï } 

• a > 0 Two real nth roots: • a > 0 One real nth root: 

6 n 

Ï } 

a 5 6a 1/n 

Example 1 Find nth Root(s) 

Find the indicated nth root(s) of a. 

n 

Ï } 

a 5 a 1/n 

a. n 5 3, a 5 227 b. n 5 6, a 5 729 

solution 

a. Because n 5 3 is odd and a 5 227 < 0, 227 

has one real cube root. Because ( 23 ) 3 5 227, 

you can write 3 Ï } 

227 5 23 . 

b. Because n 5 6 is even and a 5 729 > 0, 729 has 

two real sixth roots. Because 3 6 5 729 

and ( 23 ) 6 5 729, you can write 6 6 Ï } 

729 5 63 . 

Copyright © McDougal Littell/Houghton Mifflin Company Lesson 7.1 • Algebra 2 C&S Notetaking Guide 141


Example 2 Solve Equations Using nth Roots 

Solve the equation. 

a. 2x 6 5 1458 b. (x 2 4) 3 5 64 

Solution 

a. 2x6 5 1458 b. (x 2 4) 3 5 64 

x6 5 729 x 2 4 5 3 Ï } 

64 

Checkpoint Find the indicated nth root(s) of a. 

1. n 5 4, a 5 625 

25, 5 

Solve the equation. 

3. 3x 5 5 729 

3 

x 5 6 6 

Ï } 

729 x 2 4 5 4 

x 5 63 x 5 8 

2. n 5 3, a 5 512 

8 

4. (x 1 6) 4 5 256 

210, 22 

Example 3 Evaluate Expressions with Rational Exponents 

Evaluate the expression. 

a. 8 1/3 b. 64 1/2 

c. (21024) 1/5 d. (216) 1/4 

Solution 

a. 81/3 5 3 Ï } 

8 5 2 

b. 641/2 5 Ï } 

64 5 8 

c. (21024) 1/5 5 5 Ï } 

21024 5 24 

d. (216) 1/4 5 4 

Ï } 

216 , no real solution 

142 Lesson 7.1 • Algebra 2 C&S Notetaking Guide Copyright © McDougal Littell/Houghton Mifflin Company


Homework 

Rational ExponEnts 

Let a 1/n be an nth root of a (a > 0 and a Þ 1), and 

let m be a positive integer. 

a m/n 5 (a 1/n ) m 5 ( n 

Ï } 

a ) m 

a2m/n 5 1 

} 

a 

1 

(a1/n 5 

) m 1 

5 } 

m/n 

Example 4 Rewrite Expressions 

} 

1 n 

Ï } 

a 2 m 

a. Rewrite 1 3 

Ï } 

7 2 4 using rational exponents. 

b. Rewrite 9 23/5 using radicals. 

a. 1 3 

Ï } 

7 2 4 5 (7 1/ 3 ) 4 5 7 4/3 

b. 923/5 5 1 

9 

5. 

} 5 

3/5 1 

} 

1 91/5 5 

2 3 1 

} 

1 5 

Ï } 

9 2 3 

Example 5 Evaluate Expressions with Rational Exponents 

a. Evaluate 32 22/5 . b. Evaluate 8 4/3 . 

a. 3222/5 5 1 

} 

1 

5 } 

322/5 1 5 

Ï } 

32 2 

b. 8 4/3 5 1 3 

Ï } 

8 2 4 5 2 4 5 16 

Checkpoint Rewrite the expression. 

1 

} 

1 3 

Ï } 

7 2 8 

7 23/8 

Evaluate the expression. 

7. (2125) 22/3 

1 

} 

25 

1 1 

5 } 5 } 

2 2 4 

2 

6. 5 4/9 

1 9 

Ï } 

5 2 4 

8. 16 3/2 

Copyright © McDougal Littell/Houghton Mifflin Company Lesson 7.1 • Algebra 2 C&S Notetaking Guide 143 

64

7.2 


Properties of Rational 

Exponents 

Goal p Use properties of radicals and rational exponents. 

VocabulaRy 

Simplest form of a radical For a radical with index 

n, the form where there are no perfect nth powers 

in the radicand, and there are no radicals in any 

denominators 

Like radicals Radicals that have the same index and 

the same radicand 

PRoduct and QuotiEnt PRoPERtiEs of Radicals 

Product Property n 

Ï } 

a p b 5 n 

Ï } 

a p n 

Ï } 

b 

Ï } 

Quotient Property n a 

} 

b 5 n 

Ï } 

a 

} n 

b 

Ï } 

Example 1 Use Properties of Radicals 

Use the properties of radicals to simplify the expression. 

a. 5 

Ï } 

27 p 5 

Ï } 

9 5 5 

Ï } 

27 p 9 Product property 

b. 3 

Ï } 

192 

} 

3 

Ï } 

3 

5 3 

5 5 

Ï }} 

3 p 3 p 3 p 3 p 3 5 3 

} 

Ï 192 

} 

Ï } 

Write 5 

Ï } 

128 in simplest form. 

5 

Ï } 

128 5 5 

Ï } 

32 p 4 

5 

3 3 

64 5 4 Quotient property 

Example 2 Write Radicals in Simplest Form 

Factor out perfect fifth power. 

5 5 

Ï } 

32 p 5 

Ï } 

4 Product property of radicals 

5 2 5 

Ï } 

4 Simplify. 



ProPerties of rational exPonents 

Let a and b be positive real numbers. Let m and n 

be rational numbers. 

PROPERTY EXAMPLE 

1. am p an 5 am 1 n 41/2 p 43/2 5 4 ( 1/2 1 3/2 ) 

5 42 5 16 

2. (am ) n 5 amn (25/2 ) 2 5 2 ( 5/2 p 2 ) 

5 25 5 32 

3. (ab) m 5 a m b n (16 p 4) 1/2 5 16 1/2 p 4 1/2 

4. a2m 5 1 

} 

am , a Þ 0 2521/2 5 1 

} 

5. am 

} 

6. 1 a 

} 

5 4 p 2 5 8 

251/2 5 1 

} 

5 

an 5 am 2 n , a Þ 0 35/2 

} 

31/2 5 3( 5/2 2 1/2 ) 5 32 5 9 

b 2 m 5 am 

} 

bm , b Þ 0 1 27 

} 

8 2 1/3 5 271/3 

} 

81/3 5 3 

} 

2 

example 3 Use Properties of Rational Exponents 

Simplify the expression. 

a. 9 1/2 p 9 1/4 5 9 ( 1/2 1 1/4 ) 5 9 3/4 

b. (7 2/3 ) 3 5 7 ( 2/3 p 3 ) 5 7 2 5 49 

c. 1 16 

} 

625 2 1/4 5 161/4 

} 

d. 81 21/4 5 1 

} 

6251/4 5 2 

} 

5 

811/4 5 1 

} 

3 

e. 35/6 

} 

31/3 5 3( 5/6 2 1/3 ) 5 33/6 5 3 1/2 5 Ï } 

3 

example 4 Add or Subtract Like Radicals 

Perform the indicated operation. 

a. 6 4 

Ï } 

5 1 8 4 

Ï } 

5 5 ( 6 1 8 ) 4 

Ï } 

5 5 14 4 

Ï } 

5 

b. 12(4 3/5 ) 2 5(4 3/5 ) 5 ( 12 2 5 )4 3/5 5 7 (4 3/5 ) 



Checkpoint Simplify the expression. 

1. Ï} 245 

} 

Ï } 

5 

7 

3. 3 Ï } 

250 

5 3 

Ï } 

2 

2. 3 Ï } 

6 p 3 Ï } 

36 

6 

4. 14(4 2/3 ) 2 12(4 2/3 ) 

2(4 2/3 ) 

Example 5 Simplify Expressions with Variables 

Simplify the expression. Write the answer using positive 

exponents only. Assume all variables are positive. 

a. 5 Ï } 

32x15 5 5 

Ï } 

25 (x3 ) 5 Factor out perfect fifth powers. 

b. (36m 4 n 10 ) 1/2 

Ï 

c. 3 

5 2x 3 Simplify. 

5 36 1/2 (m 4 ) 1/2 (n 10 ) 1/2 Power of a product 

property 

5 36 1/2 m ( 4 p 1/2 ) n ( 10 p 1/2 ) Power of a power 

property 

5 6m 2 n 5 Simplify. 

} 

a9 } 5 6 

b 

5 a3 

} 

b 2 

d. 42x3/2 z 7 

} 

6x 4 y 23 z 5 

Ï} 

3 

(a3 ) 3 

} 

(b 2 ) 3 

Factor out perfect cube powers. 

Simplify. 

5 42 

} 

6 x( 3/2 2 4 ) y [0 2 (23) ] z ( 7 2 5 ) Quotient of 

powers property 

5 7x 25/2 y 3 z 2 5 7y3z2 } 

x5/2 Simplify. 



Homework 

Example 6 Add/Subtract Expressions with Variables 

Simplify the expression. Assume all variables are 

positive. 

a. 10 5 Ï } 

y 2 6 5 Ï } 

y b. 3a 2 b 1/4 1 4a 2 b 1/4 

Solution 

a. 10 5 Ï } 

y 2 6 5 Ï } 

y 5 ( 10 2 6 ) 5 Ï } 

y 

5 4 5 Ï } 

y 

b. 3a 2 b 1/4 1 4a 2 b 1/4 5 ( 3 1 4 )a 2 b 1/4 

5 7a 2 b 1/4 

Checkpoint Simplify the expression. Write your answer 

using positive exponents only. Assume all variables 

are positive. 

5. 3 Ï } 

8x 7 y 3 z 11 

2x2yz3 3 

Ï } 

xz2 6. c5 d 1/2 

} 

4cd 5/2 

c 4 

} 

4d 2 

Simplify the expression. Assume all variables are positive. 

7. 9 7 Ï } 

5y4 1 16 7 Ï } 

5y4 25 7 

Ï } 

5y4 8. 8k 5/9 2 7k 5/9 


k 5/9


7.3 Solving Radical Equations 

Goal p Solve equations that contain radicals or rational 

exponents. 

VocabulaRy 

Radical equation An equation that contains radicals 

with the variable in the radicand 

Extraneous solution An apparent solution that does 

not make the original equation true 

SolVinG Radical EquationS 

To solve a radical equation, follow these steps: 

Step 1 Isolate the radical on one side of the 

equation, if necessary. 

Step 2 Raise each side of the equation to the 

same power to eliminate the radical. 

Step 3 Solve the resulting equation using techniques 

that you learned in previous chapters. 

Step 4 Check your solution. 

Example 1 Solve a Radical Equation 

Solve Ï } 

x 1 6 5 3. 

Solution 

Ï } 

x 1 6 5 3 Write original equation. 

1 Ï } 

x 1 6 2 2 5 32 Square each side to eliminate the 

radical. 

x 1 6 5 9 Simplify. 

x 5 3 Subtract 6 from each side. 



Example 2 Solve an Equation with Two Radicals 

Solve Ï } 

2x 1 1 2 Ï } 

10 2 x 5 0. 

Solution 

Ï } 

2x 1 1 2 Ï } 

10 2 x 5 0 Original equation 

Ï } 

2x 1 1 5 Ï } 

10 2 x Add Ï } 

10 2 x 

to each side. 

1 Ï } 

2x 1 1 2 2 5 1 Ï } 

10 2 x 2 2 Square each side. 

2x 1 1 5 10 2 x Simplify. 

2x 5 9 2 x Subtract 1 

from each side. 

3x 5 9 Add x to each side. 

x 5 3 Solve for x. 

Example 3 Solve an Equation with an Extraneous Solution 

Solve x 2 2 5 Ï } 

x 1 10 . Check for extraneous solutions. 

Solution 

x 2 2 5 Ï } 

x 1 10 Original equation 

(x 2 2) 2 5 1 Ï } 

x 1 10 2 2 Square each side. 

x 2 2 4x 1 4 5 x 1 10 Expand left side; 

simplify right side. 

x 2 2 5x 2 6 5 0 Standard form 

( x 2 6 )( x 1 1 ) 5 0 Factor. 

x 2 6 5 0 or x 1 1 5 0 Zero product property 

x 5 6 or x 5 21 Solve for x. 

CHECK Substitute 6 for x and 21 for x in the original 

equation. 

6 2 2 0 Ï } 

6 1 10 21 2 2 0 Ï } 

21 1 10 

4 0 Ï } 

16 23 0 Ï } 

9 

4 5 4 ✓ 23 Þ 3 ✓ 

Because 21 does not satisfy the original equation, the 

only solution is 6 . 



Homework 

Example 4 Solve an Equation with Rational Exponents 

Solve (3x 1 4) 2/3 5 16. Check for extraneous solutions. 

Solution 

(3x 1 4) 2/3 5 16 Write original equation. 

[(3x 1 4) 2/3 ] 3/2 5 (16) 3/2 Raise each side to the 3 

} 

2 

power. 

3x 1 4 5 1 Ï } 

162 3 

3x 1 4 5 64 Simplify. 

Definition of rational 

exponents 

3x 5 60 Subtract 4 from each 

side. 

x 5 20 Divide each side by 3 . 

The solution is 20 . Check this in the original equation. 

Checkpoint Solve the equation. Check for extraneous 

solutions. 

1. 4 Ï } 

x 1 3 5 2 

13 

3. Ï } 

4x 1 5 5 3Ï } 

x 

1 

5. 22x 4/3 2 21 5 253 

28 or 8 

2. 3 Ï } 

x 2 5 1 1 5 21 

23 

4. Ï } 

3x 2 Ï } 

x 1 14 5 0 

7 

6. x 1 2 5 Ï } 

2x 1 7 

150 Lesson 7.3 • Algebra 2 C&S Notetaking Guide Copyright © McDougal Littell/Houghton Mifflin Company 

1


7.4 Function Operations and 

Composition of Functions 

Goal p Perform operations with functions, including 

composition of functions. 

VOCabulary 

Composition of functions Composing a function f 

with a function g is accomplished by replacing 

each variable in f with the expression for g 

OperatiOns On FunCtiOns 

Let f(x) and g(x) be any two functions. You can add, 

subtract, multiply, or divide f(x) and g(x) to form a new 

function h(x). 

Operation Definition Example: f(x) 5 3x, 

g(x) 5 x 1 3 

Addition h(x) 5 f(x) 1 g(x) h(x) 5 3x 1 (x 1 3) 

5 4x 1 3 

Subtraction h(x) 5 f(x) 2 g(x) h(x) 5 3x 2 (x 1 3) 

5 2x 2 3 

Multiplication h(x) 5 f(x) p g(x) h(x) 5 3x(x 1 3) 

5 3x2 1 9x 

Division h(x) 5 f(x) 

} 

g(x) 

3x 

h( x) 5 } 

x 1 3 , 

x Þ 23 

The domain of h consists of the x-values that are in the 

domains of both f and g . When h involves division, 

the domain does not include x-values for which 

the denominator is equal to zero . 



Example 1 Add and Subtract Functions 

Let f(x) 5 3x 2 1 1 and g(x) 5 x 2 5. Find h(x) and 

state its domain. 

a. h(x) 5 f(x) 1 g(x) b. h(x) 5 f(x) 2 g(x) 

Solution 

a. f(x) 1 g(x) 5 3x 2 1 1 1 (x 2 5) 

5 3x 2 1 x 2 4 

b. f(x) 2 g(x) 5 3x 2 1 1 2 (x 2 5) 

5 3x 2 2 x 1 6 

In both parts (a) and (b), the domains of f and g are 

all real numbers . So, the domain of h is 

all real numbers . 

Example 2 Multiply and Divide Functions 

Let f(x) 5 x3 and g(x) 5 7x. Find h(x) and state its 

domain. 

a. h(x) 5 f(x) p g(x) b. h(x) 5 f(x) 

} 

g(x) 

Solution 

a. f(x) p g(x) 5 (x 3 )(7x) 

b. 

5 7x 4 

The domains of both f and g are all real numbers . 

So, the domain of h(x) is all real numbers . 

f(x) 

} 

g(x) 

5 x3 

} 

7x 

5 1 

} x ( 3 2 1 ) 

7 

5 1 

} 

7 x 2 

Because g(0) 5 0 , f 

} 

g is undefined when x 5 0 . 

So, the domain of h(x) is all real numbers 

except 0 . 



Checkpoint Complete the following exercise. 

1. Let f(x) 5 5x3 and g(x) 5 22x3 . Find (a) f 1 g, 

(b) f 2 g, (c) f p g, (d) f 

} , and (e) the domains. 

g 

a. 3x3 b. 7x3 c. 210x6 d. 2 1 

} 

7 

e. The domain of f 1 g, f 2 g, and f p g is all real 

numbers. The domain of f 

} 

g 

is all real numbers 

except 0. 

Composition of funCtions 

The composition of a function f with a function g is: 

h(x) 5 g(f (x)) 

The domain of h is the set of all x-values where x is 

in the domain of g and g(x) is in the domain of f . 

Domain of g Range of g 

Input 

of g 

x 

Output 

of g 

g(x) 

Input 

of f 

f(g(x)) 

Output 

of f 

Domain of f Range of f 



Homework 

Example 3 Write a Composition of Functions 

Let f (x) 5 6x and g(x) 5 3x 1 5. Find the following. 

a. f (g(x)) b. g(f (x)) 

c. the domain of each composition 

Solution 

To find the compositions, write the composition, substitute 

the expression for the inner function in the outer 

function, and simplify. 

a. f (g(x)) 5 f( 3x 1 5 ) 5 6( 3x 1 5 ) 5 18x 1 30 

b. g(f (x)) 5 g( 3x ) 5 3( 3x ) 1 5 5 18x 1 5 

c. The domain of each function is all real numbers , so 

the domain of each composition is all real numbers . 

Example 4 Evaluate a Composition of Functions 

Let f(x) 5 6x and g(x) 5 3x 1 5. Evaluate f(g(22)). 

Solution 

To evaluate f(g(22)), first find g(22): 

g(22) 5 3( 22 ) 1 5 5 26 1 5 5 21 

Then substitute g(22) 5 21 into f (g(22)): 

f(g(22)) 5 f( 21 ) 5 6( 21 ) 5 26 


2. Let f (x) 5 5x 2 4 and g(x) 5 3x. Find 

(a) f (g(x)), (b) g(f (x)), (c) f (g(23)), and (d) g(f (4)). 

a. 15x 2 4 b. 15x 2 12 c. 249 d. 48 


7.5 


Inverse Functions 

Goal p Find the inverses of linear and nonlinear functions. 

Vocabulary 

Inverse relation A relation that switches the input 

and output values of the original relation 

Inverse functions If a relation and its inverse are 

functions, then the two functions are inverse 

functions. 

InVerse FunctIons 

Functions f and g are inverses if: 

f(g(x)) 5 x and g(f (x)) 5 x 

The inverse of f is denoted by f –1 , which is read as 

“f inverse .” 

example 1 Verify Inverse Functions 

Verify that f(x) 5 7x 2 4 and g(x) 5 1 

} x 1 

7 4 

} are inverse 

7 

functions. 

solution 

To verify f(x) and g(x) are inverse functions, show that 

f(g(x)) 5 x and g(f(x)) 5 x . 

f(g(x)) 5 f 1 1 

} x 1 

7 

4 

} 

7 2 

g(f(x)) 5 g(7x 2 4) 

5 7 1 1 

} x 1 

7 4 

1 

} 

7 2 2 4 5 } ( 7x 2 4 ) 1 

7 4 

} 

7 

5 x 1 4 2 4 5 x 2 4 

} 1 

7 4 

} 

7 

5 x ✓ 5 x ✓ 



Finding inverse Functions 

You can find the inverse of a function by following these 

steps. 

Step 1 Replace f(x) with y (if the function is written 

using function notation). 

Step 2 Switch x and y. 

Step 3 Solve for y . 

example 2 Find the Inverse of a Linear Function 

Find the inverse of the function f(x) 5 23x 1 6. 

solution 

f(x) 5 23x 1 6 Write original function. 

y 5 23x 1 6 Replace f(x) with y. 

x 5 23y 1 6 Switch x and y. 

x 2 6 5 23y Subtract 6 from each side. 

2 1 

} x 1 2 5 y 

3 

Divide each side by –3 to 

solve for y . 

The inverse function is f 21 (x) 5 2 1 

} 

3 x 1 2 . 


1. Find the inverse of the function g(x) 5 7x 2 3. Then 

verify that the two functions are inverses of each 

other. 

g –1 x 1 3 

(x) 5 } 

7 ; 

g(g –1 x 1 3 

(x)) 5 7 1 } 

7 2 2 3 5 x 1 3 2 3 5 x and 

g –1 (7x 2 3) 1 3 7x 

(g(x)) 5 }} 5 } 5 x 

7 7 



Homework 

Horizontal line test 

The inverse of a function f is also a function if and 

only if no horizontal line intersects the graph of 

f more than once . 

example 3 Find the Inverse of a Nonlinear Function 

Determine whether the inverse of f(x) 5 1 

} 

4 x 3 1 3 is a 

function. Then find the inverse. 

solution 

First, graph the function on a graphing calculator. Because 

no horizontal line intersects the graph more than once, 

the inverse of f is also a function . 

Next, find an equation for f21 . 

f(x) 5 1 

} x 

4 3 1 3 Write original function. 

Checkpoint Determine whether the inverse of f is a 

function. Then find the inverse. 

2. f(x) 5 2x 4 1 1, x ≥ 0 

y 5 1 

} 

4 x 3 1 3 Replace f(x) with y. 

x 5 1 

} 

4 y 3 1 3 Switch x and y. 

x 2 3 5 1 

} 

4 y 3 Subtract 3 from each side. 

4x 2 12 5 y3 Multiply each side by 4 . 

3 

Ï } 

4x 2 12 5 y Take the cube root of 

each side. 

yes; f21 ( x) 5 4 

} 

x 2 1 

} Ï 

x ≥ 1 

2 , 

3. f(x) 5 1 

} 

3 x 2 2 5, x ≥ 0 

yes; f –1 ( x) 5 Ï } 

3x 1 15 , 

x ≥ 25 


7.6 


Graphing Square Root and 

Cube Root Functions 

Goal p Graph square root and cube root functions. 

VoCabulaRy 

Radical function A function that contains a radical 

with a variable in its radicand 

GRaphS oF SquaRe Root/Cube Root FunCtionS 

Square Root Function Cube Root Function 

(0, 0) 

y 

y 5 a x 

(1, a) 

x 

(21, 2a) 

y 

3 y 5 a x 

(1, a) 

(0, 0) 

The domain of y 5 aÏ } 

x 

is x ≥ 0. The range is 

The domain and range 

of y 5 a 3 Ï } 

x are 

y ≥ 0 when a > 0. all real numbers . 

example 1 Graph a Square Root Function 

Graph y 5 2 Ï } 

x . Then state its domain and range. 

Make a table of values. Then plot the points from the 

table and connect them with a smooth curve. 

x 0 1 2 3 4 

y 0 2 2.83 3.46 4 

1 

O 

y 

1 

y 5 2 x 

The radicand of a square root is always nonnegative , 

so the domain is x ≥ 0. The range is y ≥ 0 . 


x 

x


Example 2 Graph a Cube Root Function 

Graph y 5 2 1 

} 

2 3 

Ï } 

x . Then state its domain and range. 

x –2 –1 0 1 2 

y 0.63 0.5 0 –0.5 –0.63 

y 5 2 3 1 

2 x 

1 

O 

y 

1 

x 

The radicand of a cube root can 

be any real number, so the 

domain is all real numbers . 

You can see from the graph that 

the range is all real numbers . 

Checkpoint Graph the function. Then state its domain 

and range. 

1. y 5 2 3 

Ï } 

x 2. y 5 22 Ï } 

x 

1 

O 

y 

1 

y 5 2 3 x 

x 

Graphs of radical functions 

To graph y 5 a Ï } 

x 2 h 1 k or y 5 a 3 

Ï } 

2 h 1 k: 

Step 1 Sketch the graph of y 5 a Ï } 

x or y 5 a 3 

Ï } 

x . 

Step 2 Determine the values of h and k. Translate the 

graph h units horizontally and k units 

vertically . If h > 0, translate the graph 

to the right ; if h < 0, translate it to the left . 

If k > 0, translate the graph up ; if k < 0, 

translate it down . 

O 

22 

y 

1 

y 5 22 x 

domain and range: domain: x ≥ 0, 

all real numbers range: y ≤ 0 


x


Homework 

Example 3 Graph a Square Root Function 

Graph y 5 Ï } 

x 2 1 1 2. State its domain and range. 

1. Sketch the graph of y 5 Ï } 

x . 

2. Determine the values of h 

and k. Because h 5 1 , you 

translate the graph of y 5 Ï } 

x 

1 unit right . Because 

k 5 2 , you translate the 

graph 2 units up . 

The domain is x ≥ 1 . The range is y ≥ 2 . 

Checkpoint Graph the function. State domain and range. 

3. y 5 Ï } 

x 1 3 1 2 4. y 5 3 Ï } 

x 1 1 

1 

O 

y 

y 5 x 1 3 1 2 

1 

x 

1 

O 

y 

y 5 3 x 1 1 

domain: x ≥ 23, domain and range: 

range: y ≥ 2 all real numbers 

1 

O 

y 

1 

1 

y 5 x 2 1 1 2 

Graph y 5 3 

Ï } 

Example 4 Graph a Cube Root Function 

x 1 3 2 2. State its domain and range. 

1. Sketch the graph of y 5 3 Ï } 

x . 

2. Determine the values of h and 

k. Because h 5 23 , you 

translate the graph of y 5 3 Ï } 

x 

3 units left . Because 

k 5 22 , you translate the 

graph 2 units down . 

The domain and range are all real numbers . 

1 

y 5 3 x 1 3 2 2 


y 

1 

x 

x 

x

7.7 


Standard Deviation 

Goal p Use standard deviation to describe data sets. 

Vocabulary 

Standard deviation The typical difference (or 

deviation) between the mean and any data value 

in a set 

StanDarD DeViation of a Set of Data 

To find the standard deviation of x 1 , x 2 , x 3 , …, x n : 

Step 1 Find the mean of the data set, } 

x (read as “x bar”). 

Step 2 Find the standard deviation s (read as “sigma”). 

s 5 Ï }}}} 

(x1 2 } x ) 2 1 (x2 2 } x ) 2 1 . . . 1 (xn 2 } 

x ) 2 

}}}} 

n 

example 1 Find Standard Deviation 

The height, in inches, of six trees at a nursery are 

shown. Find the standard deviation of the heights. 

36 48 50 44 53 39 

Solution 

1. Find the mean. 

} 

x 5 

36 1 48 1 50 1 44 1 53 1 39 

}}} 5 45 

6 

2. Find the standard deviation. 

s 5 

Ï }}}}} 

(36245)2 1 (48245) 2 1 (50245) 2 1 (44245) 2 1 (53245) 2 1 (39245) 2 

}}}}}} 

6 

} 

5 Ï 216 

} 

6 

5 6 

The standard deviation is 6 . 



Example 2 Compare Data After Adding a Constant 

Suppose that 6 months later, each of the trees in 

Example 1 had grown 5 inches. 

a. Find the standard deviation of the heights after 

6 months. 

b. Compare this standard deviation with the standard 

deviation of the trees found in Example 1. 

c. Make a box-and-whisker plot of the data used in 

Example 1 and the new data. What conclusions can 

you make about the two data sets? 

Solution 

a. } 

x 5 

s 5 

41 1 53 1 55 1 49 1 58 1 44 

}}} 5 50 

6 

Ï }}}}} 

(41250) 2 1 (53250) 2 1 (55250) 2 1 (49250) 2 1 (58250) 2 1 (44250) 2 

}}}}}} 

6 

} 

5 Ï 216 

} 5 6 

6 

b. The standard deviation is the same as in Example 1. 

c. The two data sets have exactly the same shape. 

The graph for the new data is 5 units to the right 

of the graph for the data in Example 1. 

30 

35 

40 45 

36 39 46 50 53 

50 55 60 

41 44 51 55 58 

Checkpoint Complete the exercise. 

1. The speeds, in miles per hour, of five different cars on 

a local highway are 69, 62, 64, 60, and 65. (a) Find the 

standard deviation of the speeds. (b) Find the standard 

deviation of the speeds if the cars are driven 10 miles 

per hour slower. (c) Compare the standard deviations. 

a. 4.6 b. 4.6 c. The standard deviations are the 

same. 



Homework 

Example 3 Compare Data Sets 

The data sets below give the quiz scores for the 

students in two different biology classes. 

Class A: 15, 17, 17, 17, 18, 19, 21, 22, 25 

Class B: 16, 18, 19, 21, 22, 22, 22, 24, 25 

a. Find the mean and standard deviation for each class. 

b. Compare the mean and standard deviation of the two 

classes. 

c. What conclusions can you draw? 

Solution 

a. Class A: Mean 5 19 

Standard deviation < 2.94 

Class B: Mean 5 21 

Standard deviation < 2.71 

b. The mean for Class A is less than the mean for 

Class B, but its standard deviation is greater than 

the standard deviation for Class B. 

c. Since the standard deviation for Class A is greater 

than the standard deviation for Class B, the scores for 

Class A are more spread out than the scores for 

Class B. 

Checkpoint Tell which data set is more spread out. 

2. Set A: } x 5 63, s 5 6.7 

Set B: } x 5 58, s 5 5.98 

Set A 

3. Set A: } x 5 115, s 5 1.25 

Set B: } x 5 95, s 5 1.25 

equally spread out 


Words to Review 

Use your own words and/or an example to explain 

the vocabulary word. 

nth root of a real number 

For an integer n greater 

than 1, if bn 5 a, then 

b is the nth root of a 

(Written as n 

a .) 

Ï } 

Simplest form of a radical 

The radical 4 

Ï } 

2 

} is in 

2 

simplest form. 

Radical equation 

An equation that 

contains radicals with 

the variable in the 

radicand 

Composition of functions 

If f (x) 5 3x and 

g(x) 5 2x 2 1, then 

f (g(x)) 5 3(2x 2 1) 5 

6x 2 3 and g(f (x)) 5 

2(3x) 2 1 5 6x 2 1. 

Inverse functions 

f(x) 5 x 1 6 and 

g(x) 5 x 2 6 

Standard deviation 

Index 

The index of 3 

Ï } 

8 is 3. 

Like radicals 

3 

Ï } 

2 and 4 3 

Ï } 

2 

Extraneous solution 

An apparent solution 

that does not make the 

original equation true 

Inverse relation 

A relation that 

switches the input and 

output values of the 

original relation 

Radical function 

g(x) 5 Ï } 

3x 1 1 

The typical difference (or deviation) between the 

mean and any data value in a set 

Review your notes and Chapter 7 by using the 

Chapter Review on pages 401–404 of your textbook. 

164 Words to Review • Algebra 2 C&S Notetaking Guide Copyright © McDougal Littell/Houghton Mifflin Company

Notes with answers

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