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Chapter 6 Review6-1 - Operations on FunctionsArithmetic Operations

Operations with Functions

Sum (f + g)(x) = f (x) + g(x)

Difference (f – g)(x) = f (x) – g(x)

Product (f ⋅ g)(x) = f (x) ⋅ g(x)

Quotient ( fg )(x) =

f (x )g (x)

, g(x) ≠ 0

Example : Find (f + g)(x), (f – g)(x), (f ⋅ g)(x), and ( fg ) (x) for f(x) = x2 + 3x – 4 and g(x) = 3x – 2.

(f + g)(x) = f(x) + g(x) Addition of functions

= (x2 + 3x – 4) + (3x – 2) f(x) = x2 + 3x – 4, g(x) = 3x – 2

= x2 + 6x – 6 Simplify.

(f – g)(x) = f(x) – g(x) Subtraction of functions

= (x2 + 3x – 4) – (3x – 2) f(x) = x2 + 3x – 4, g(x) = 3x – 2

= x2 – 2 Simplify.

(f ⋅ g)(x) = f(x) ⋅ g(x) Multiplication of functions

= (x2 + 3x – 4)(3x – 2) f(x) = x2+ 3x – 4, g(x) = 3x – 2

= x2(3x – 2) + 3x(3x – 2) – 4(3x – 2) Distributive Property

= 3x3 – 2x2 + 9x2 – 6x – 12x + 8 Distributive Property

= 3x3 + 7x2 – 18x + 8 Simplify.

( fg )(x) =

f (x )g (x)

Division of functions

= x2+3 x−43 x−2

, x ≠ 23

f(x) = x2 + 3x – 4 and g(x) = 3x – 2

Exercises

Find (f + g)(x), (f – g)(x), (f ⋅ g)(x), and ( fg ) (x) for each f(x) and g(x).

1. f(x) = 8x – 3; g(x) = 4x + 5 2. f(x) = x2 + x – 6; g(x) = x – 2

3. f(x) = 3x2 – x + 5; g(x) = 2x – 3 4. f(x) = 2x – 1; g(x) = 3x2 + 11x – 4

6-1 Continued - Operations on Functions

Composition of Functions Suppose f and g are functions such that the range of g is a subset of the domain of f. Then the composite function f ◦ g can be described by the equation [f ° g](x) = f[g(x)].

Example 1: For f = {(1, 2), (3, 3), (2, 4), (4, 1)} and g = {(1, 3), (3, 4), (2, 2), (4, 1)}, find f ◦ g and g ◦ f if they exist.

f[ g(1)] = f(3) = 3 f[ g(2)] = f(2) = 4 f[ g(3)] = f(4) = 1 f[ g(4)] = f(1) = 2,

So f ◦ g = {(1, 3), (2, 4), (3, 1), (4, 2)}

g[ f(1)] = g(2) = 2 g[ f(2)] = g(4) = 1 g[ f(3)] = g(3) = 4 g[ f(4)] = g(1) = 3,

So g ◦ f = {(1, 2), (2, 1), (3, 4), (4, 3)}

Example2: Find [g ◦ h](x) and [h ◦ g](x) for g(x) = 3x – 4 and h(x) = x2 – 1.

[g ◦ h](x) = g[h(x)] [h ◦ g](x) = h[ g(x)]

= g(x2 – 1) = h(3x – 4)

= 3(x2 – 1) – 4 = (3 x−4)2 – 1

= 3x2 – 7 = 9x2 – 24x + 16 – 1

= 9x2 – 24x + 15

ExercisesFor each pair of functions, find f ◦ g and g ◦ f , if they exist.

1. f = {(–1, 2), (5, 6), (0, 9)}, 2. f = {(5, –2), (9, 8), (–4, 3), (0, 4)},

g = {(6, 0), (2, –1), (9, 5)} g = {(3, 7), (–2, 6), (4, –2), (8, 10)}

Find [f ◦ g](x) and [g ◦ f ](x), if they exist.

3. f(x) = 2x + 7; g(x) = –5x – 1 4. f(x) = x2 – 1; g(x) = –4x2

5. f(x) = x2 + 2x; g(x) = x – 9 6. f(x) = 5x + 4; g(x) = 3 – x

6-2 Study Guide - Inverse Functions and Relations

Find Inverses

Inverse Relations Two relations are inverse relations if and only if whenever one relation contains the element (a, b), the other relation contains the element (b, a).

Property of Inverse Functions

Suppose f and f−1 are inverse functions.

Then f (a) = b if and only if f−1(b) = a.

Example: Find the inverse of the function f(x) = 25

x – 15

. Then graph the function and its inverse.

Step 1 Replace f(x) with y in the original equation.

f(x) = 25

x – 15

→ y = 25

x – 15

Step 2 Interchange x and y.

x = 25

y – 15

Step 3 Solve for y.

x = 25

y – 15

Inverse of y = 25

x – 15

5x = 2y – 1 Multiply each side by 5.

5x + 1 = 2y Add 1 to each side.

12 (5x + 1) = y Divide each side by 2.

The inverse of f(x) = 25

x – 15

is f−1(x) = 12

(5x + 1).

ExercisesFind the inverse of each function. Then graph the function and its inverse.

1. f(x) = 23 x – 1 2. f(x) = 2x – 3 3. f(x) =

14 x – 2

6-2 - Inverse Functions and Relations (continued)Verifying Inverses

Inverse Functions Two functions f(x) and g(x) are inverse functions if and only if [f ◦ g](x) = x and [g ◦ f ](x) = x.

Example 1: Determine whether f(x) = 2x – 7 and g(x) = 12

(x + 7) are inverse functions.

[ f ◦ g](x) = f[ g(x)] [g ◦ f ](x) = g[ f(x)]

= f [12(x+7)] = g (2x – 7)

= 2 [12(x+7)] – 7 =

12

(2x – 7 + 7)

= x + 7 – 7 = x

= x

The functions are inverses since both [ f ◦ g](x) = x and [ g ◦ f ](x) = x.

Example 2: Determine whether f(x) = 4x + 13

and g(x) = 14

x – 3 are inverse functions.

[ f ◦ g](x) = f[ g(x )]

= f ( 14

x – 3)= 4 ( 1

4x – 3) +

13

= x – 12 + 13

= x – 11 23

Since [ f ◦ g](x) ≠ x, the functions are not inverses.

ExercisesDetermine whether each pair of functions are inverse functions. Write yes or no.

1. f(x) = 3x – 1 2. f(x) = 14 x + 5 3. f(x) =

12 x – 10

g(x) = 13 x +

13 g(x) = 4x – 20 g(x) = 2x +

110

4. f(x) = 2x + 5 5. f(x) = 8x – 12 6. f(x) = –2x + 3

g(x) = 5x + 2 g(x) = 18 x + 12 g(x) = –

12 +

32

6-3 - Square Root Functions and Inequalities

Square Root Functions A function that contains the square root of a variable expression is a square root function. The domain of a square root function is those values for which the radicand is greater than or equal to 0.

Example: Graph y = √3 x−2. State its domain and range.

Since the radicand cannot be negative, the domain of the function is 3x – 2 ≥ 0 or x ≥ 23

.

The x-intercept is 23

. The range is y ≥ 0.

Make a table of values and graph the function.

x y

23

0

1 1

2 2

3 √7

ExercisesGraph each function. State the domain and range.

1. y = √2 x 2. y = –3 √ x 3. y = – √ x2

4. y = 2 √ x−3 5. y = – √2x−3 6. y = √2 x+5

6-3 - Square Root Functions and Inequalities(continued)

Square Root Inequalities A square root inequality is an inequality that contains the square root of a variable expression. Use what you know about graphing square root functions and graphing inequalities to graph square root inequalities.

Example: Graph y ≤ √2 x−1 + 2.Graph the related equation y = √2x−1 + 2. Since the boundary should be included, the graph should be solid.

The domain includes values for x ≥ 12

, so the graph is to the right of x = 12

.

ExercisesGraph each inequality.

1. y < 2√ x 2. y > √ x+3 3. y < 3√2x−1

4. y < √3 x−4 5. y ≥ √ x+1 – 4 6. y > 2√2x−3

6-4 - nth Roots

Simplify Radicals

Square Root For any real numbers a and b, if a2 = b, then a is a square root of b.

nth Root For any real numbers a and b, and any positive integer n, if an = b, then a is an nth root of b.

Real nth Roots of b,n√b, – n√b

1. If n is even and b > 0, then b has one positive real root and one real negative root.

2. If n is odd and b > 0, then b has one positive real root.

3. If n is even and b < 0, then b has no real roots.

4. If n is odd and b < 0, then b has one negative real root.

Example 1: Simplify √49 z8.

√49 Z8 = √¿¿ = 7 z4

z4 must be positive, so there is no need to take the absolute value.

Example 2: Simplify – 3√(2 a−1)6

– 3√ (2 a−1 )6= 3√[ (2 a−1 )¿¿2]3¿ = – (2 a−1 )2

ExercisesSimplify.

1.√81 2.3√−343 3. √144 p6

4.±√4 a10 5.5√243 p10 6. – 3√m6n9

7.3√−b12 8.√16 a10 b8 9. √121 x6

10.√(4 k )4 11. ±√169 r4 12. – 3√−27 p6

13. – √625 y2 z4 14. √36 q34 15. √100 x2 y4 z2

6-4 - nth Roots(continued)

Approximate Radicals with a Calculator

Irrational Number a number that cannot be expressed as a terminating or a repeating decimal

Radicals such as √2 and √3 are examples of irrational numbers. Decimal approximations for irrational numbers are often used in applications. These approximations can be easily found with a calculator.

Example: Use a calculator to approximate 5√18 .2 to three decimal places.

3√18.2 ≈ 1.787

ExercisesUse a calculator to approximate each value to three decimal places.

1. √62 2. √1050 3. 3√0.054

4. – 4√5.45 5. √5280 6. √18,600

7. √0.095 8. 3√−15 9.5√100

10. LAW ENFORCEMENT The formula r = 2 √5 L is used by police to estimate the speed r in miles per hour of a car if the length L of the car’s skid mark is measures in feet. Estimate to the nearest tenth of a mile per hour the speed of a car that leaves a skid mark 300 feet long.

11. SPACE TRAVEL The distance to the horizon d miles from a satellite orbiting h miles above Earth can be approximated by d = √8000 h+h2. What is the distance to the horizon if a satellite is orbiting 150 miles above Earth?

6-5 - Operations with Radical Expressions

Simplify Radicals

Product Property of Radicals

For any real numbers a and b, and any integer n > 1:1. if n is even and a and b are both nonnegative, then n√ab = n√a ⋅ n√b.2. if n is odd, then n√ab = n√a ⋅ n√b.

To simplify a square root, follow these steps:

1. Factor the radicand into as many squares as possible.

2. Use the Product Property to isolate the perfect squares.

3. Simplify each radical.

Quotient Property of Radicals

For any real numbers a and b ≠ 0, and any integer n > 1,

n√ ab

= n√an√b

, if all roots are defined.

To eliminate radicals from a denominator or fractions from a radicand, multiply the numerator and denominator by a quantity so that the radicand has an exact root.

Example 1: Simplify 3√−6 a5 b7.

3√−16 a5 b7 = √(−2)3 ⋅2⋅ a3⋅ a2⋅(b2)3⋅b= – 2ab2 3√2 a2 b

Example 2: Simplify √ 8 x3

45 y5

√ 8 x3

45 y5 = √ 8 x3

45 y5Quotient Property

= √(2 x )2⋅2 x

√(3 y2)2 ⋅5 y Factor into squares.

= √(2x )2⋅√2 x

√(3 y2)2 ⋅√5 y Product Property

= 2|x|√2x3 y2√5 y

Simplify.

= 2|x|√2x3 y2√5 y

⋅ √5 y√5 y

Rationalize the denominator.

= 2|x|√10 xy

15 y3 Simplify.

ExercisesSimplify.

1. 5√54 2.4√32 a9 b20 3. √75 x 4 y7

4.√ 36125

5.√ a6 b3

986. 3√ p5 q3

40

6-5 - Operations with Radical Expressions(continued)

Operations with Radicals When you add expressions containing radicals, you can add only like terms or like radical expressions. Two radical expressions are called like radical expressions if both the indices and the radicands are alike.

To multiply radicals, use the Product and Quotient Properties. For products of the form (a√b + c √d) ⋅ (e √ f + g √h), use the FOIL method.

Example 1: Simplify 2√50 + 4 √500 – 6√125.

2√50 + 4√500 – 6√125 = 2√52⋅2 + 4√102⋅5 – 6√52⋅5 Factor using squares.

= 2 ⋅ 5 ⋅ √2 + 4 ⋅ 10 ⋅ √5 – 6 ⋅ 5 ⋅ √5 Simplify square roots.

= 10√2 + 40 + √5 – 30√5 Multiply.

= 10√2 + 10√5 Combine like radicals.

Example 2: Simplify (2√3 – 4√2 ) (√3+ 2√2) .

(2√3 – 4√2 ) (√3 + 2√2)

= 2 √3 ⋅√3 + 2√3 ⋅ 2√2 – 4√2⋅√3 – 4√2 ⋅ 2√2= 6 + 4√6 – 4√6 – 16

= –10

Example 3: Simplify 2−√53+√5

.

2−√53+√5

= 2−√53+√5

⋅ 3−√53−√5

= 6−2√5−3√5+(√5)2

32−(√5)2

= 6−5√5+5

9−5

= 11−5√5

4

ExercisesSimplify.

1. 3√2 + √50 – 4√8 2. √20 + √125 – √45 3. √300 – √27 – √75

4.3√81 ⋅3√24 5. 3√2 ( 3√4+ 3√12 ) 6. 2√3 (√15 + √60)

7. (2 + 3√7) (4 + √7) 8. (6√3 – 4√2) (3√3 + √2) 9. (4√2 – 3√5) (2√20 + 5)

6-6 - Rational Exponents

Rational Exponents and Radicals

Definition of b1n For any real number b and any positive integer n, b

1n = n√b, except when b < 0 and n is even.

Definition of bmn For any nonzero real number b, and any integers m and n, with n > 1, b

mn = n√bm = ( n√b )m, except when b < 0 and

n is even.

Example 1: Write 2812 in radical form.

Notice that 28 > 0.

2812 = √28

= √22⋅7= √22 ⋅ √7

= 2√7

Example 2: Evaluate ( −8−125 )

13 .

Notice that –8 < 0, –125 < 0, and 3 is odd.

( −8−125 )

13 =

3√−83√−125

= −2−5

= 25

ExercisesWrite each expression in radical form, or write each radical in exponential form.

1. 1117 2. 15

13 3. 300

32

4. √47 5. 3√3a5 b2 6. 4√162 p5

Evaluate each expression.

7. −2723 8. 216

13 9. (0.0004)

12

6-6 - Rational Exponents(continued)

Simplify Expressions All the properties of powers from Lesson 6-1 apply to rational exponents. When you simplify expressions with rational exponents, leave the exponent in rational form, and write the expression with all positive exponents. Any exponents in the denominator must be positive integers.

When you simplify radical expressions, you may use rational exponents to simplify, but your answer should be in radical form. Use the smallest index possible.

Example 1: Simplify y23 ⋅ y 3

8 .

y23 ⋅ y 3

8 = y23+3

8 = y2524

Example 2: Simplify 4√144 x6.

4√144 x6 = (144 x6 )14

= (24 ⋅32⋅ x6 )14

= (24 )14 ⋅ (32 )

14 ⋅ ( x6 )

14

= 2 ⋅ 312 ⋅ x 3

2 = 2x ⋅ (3 x )12 = 2x√3 x

ExercisesSimplify each expression.

1. x45 ⋅ x 6

5 2. ( y23 )

34 3. p

45 ⋅ p

710

4. (m65 )

25 5. x

38 ⋅ x 4

3 6. (s 16)

43

7. p

p13

8. x

12

x13

9. 6√128 SKIP

10. 4√49 SKIP 11. a 3√b4

√ab3 12. √32 ⋅ 3√16 TRY

6-7 - Solving Radical Equations and Inequalities

Solve Radical Equations The following steps are used in solving equations that have variables in the radicand. Some algebraic procedures may be needed before you use these steps.

Step 1 Isolate the radical on one side of the equation.Step 2 To eliminate the radical, raise each side of the equation to a power equal to the index of the radical.Step 3 Solve the resulting equation.Step 4 Check your solution in the original equation to make sure that you have not obtained any extraneous roots.

Example 1: Solve 2√4 x+8 – 4 = 8.

2√4 x+8 – 4 = 8 Original equation

2√4 x+8 = 12 Add 4 to each side.

√4 x+8 = 6 Isolate the radical.

4x + 8 = 36 Square each side.

4x = 28 Subtract 8 from each side.

x = 7 Divide each side by 4.

Check2√4 (7)+8 – 4 ≟ 8

2√36 – 4 ≟ 82(6) – 4 ≟ 8

8 = 8The solution x = 7 checks.

Example 2: Solve √3 x+1 = √5 x – 1.

√3 x+1 = √5 x – 1 Original equation

3x + 1 = 5x – 2√5 x + 1 Square each side.

2√5 x = 2x Simplify.

√5 x = x Isolate the radical.

5x = x2 Square each side.

x2 – 5x = 0 Subtract 5x from each side.

x(x – 5) = 0 Factor.

x = 0 or x = 5

Check

√3(0)+1 = 1, but √5(0) – 1 = –1, so 0 is not a solution.

√3(5)+1 = 4, and √5(5) – 1 = 4, so the solution is x = 5.

ExercisesSolve each equation.

1. 3 + 2x√3 = 5 2. 2√3 x+4 + 1 = 15 3. 8 + √ x+1 = 2

4. √5−x – 4 = 6 5. 12 + √2x−1 = 4 6. (9 x−11)12 = x + 1

7. 43√2x+11 – 2 = 10 8. 10 – √2x = 5 9. √4+7 x = √7 x−9

6-7 - Solving Radical Equations and Inequalities(continued)

Solve Radical Inequalities A radical inequality is an inequality that has a variable in a radicand. Use the following steps to solve radical inequalities.

Step 1 If the index of the root is even, identify the values of the variable for which the radicand is nonnegative.Step 2 Solve the inequality algebraically.Step 3 Test values to check your solution.

Example: Solve 5 – √20 x+4 ≥ –3.

Since the radicand of a square root must be greater than or equal to zero, first solve20x + 4 ≥ 0.20x + 4 ≥ 0

20x ≥ –4

x ≥ – 15

Now solve 5 – √20 x+4 ≥ – 3.

5 – √20 x+4 ≥ –3 Original inequality

√20 x+4 ≤ 8 Isolate the radical.

20x + 4 ≤ 64 Eliminate the radical by squaring each side.

20x ≤ 60 Subtract 4 from each side.

x ≤ 3 Divide each side by 20.

ExercisesSolve each inequality.

1. √c−2 + 4 ≥ 7 2. 3√2x−1 + 6 < 15 3. √10 x+9 – 2 > 5

4. 8 – √3 x+4 ≥ 3 5. √2x+8 – 4 > 2 6. 9 – √6 x+3 ≥ 6