Beginning and Intermediate Algebra - Wallace Math Courses ...

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Example 524. Are f(x)= 3√ 3x +4 and g(x) = x3 −4 inverses? Caculate composition 3 f(g(x)) Replace g(x) with x3 − 4 � � � �3 3 3 x − 4 x − 4 f Substitute for variable in f 3 3 � � � 3 3 x − 4 3 +4 Divide out the 3 3 ′ s x3 3√ − 4+4 Combine like terms x3 3√ Take cubed root Example 525. x Simplified to x! Yes, they are inverses! Our Solution Are h(x) =2x +5and g(x)= x − 5 inverses? 2 h(g(x)) � x � h − 5 � 2 x � 2 − 5 +5 2 Calculate composition � x � Replace g(x) with − 5 � 2 x � Substitute − 5 for variable inh 2 Distrubte 2 x − 10 +5 Combine like terms Example 526. Are f(x) = x − 5 Did not simplify to x No, they are not inverses Our Solution 3x −2 x +2 and g(x) = inverses? Calculate composition 4x +1 3 − 4x � � x +2 f(g(x)) Replace g(x) with 3 − 4x � � � � x +2 x + 2 f Substitute for variable in f 3 − 4x 3 − 4x � x + 2 3 4 3 − 4x � x + 2 3 − 4x � − 2 � + 1 402 Distribute3and 4 into numerators
(3x +6)(3 − 4x) 3 − 4x (4x+8)(3 − 4x) 3 − 4x 3x + 6 3 − 4x 4x+8 3 − 4x − 2 − 2(3 − 4x) +1(3 − 4x) + 1 Multiply each term by LCD:3−4x Reduce fractions 3x + 6 − 2(3 − 4x) 4x + 8+1(3 − 4x) Distribute 3x + 6 − 6+8x 4x + 8+3−4x 11x 11 Combine like terms Divide out 11 x Simplified to x! Yes, they are inverses Our Solution While the composition is useful to show two functions are inverses, a more common problem is to find the inverse of a function. If we think of x as our input and y as our output from a function, then the inverse will take y as an input and give x as the output. This means if we switch x and y in our function we will find the inverse! This process is called the switch and solve strategy. Switch and solve strategy to find an inverse: 1. Replace f(x) with y 2. Switch x and y’s 3. Solve for y 4. Replace y with f −1 (x) Example 527. Find the inverse of f(x) =(x + 4) 3 − 2 Replace f(x) with y y =(x + 4) 3 − 2 Switch x and y x = (y + 4) 3 − 2 Solve for y + 2 +2 Add 2 to both sides 403
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Beginning and Intermediate Algebra
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Special thanks to: My beautiful wif
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Chapter 6: Factoring 6.1 Greatest C
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0.1 Pre-Algebra - Integers Objectiv
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Multiplication and division of inte
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45) (4)( − 6) Find each quotient.
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9 ÷ 3 3 = 21 ÷ 3 7 Our Soultion T
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Example 21. 13 6 − 9 6 4 6 2 3 Sa
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Find each quotient. 37) − 2 ÷ 7
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the expression that is to be evalua
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Solve. 1) − 6 · 4( − 1) 3) 3+(
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It will be more common in our study
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0.4 Practice - Properties of Algebr
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Chapter 1 : Solving Linear Equation
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Addition Problems To solve equation
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8x = − 24 8 8 x = − 3 Division
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1.2 Linear Equations - Two-Step Equ
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− 1 − 1 Divide both sides by
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1.3 Solving Linear Equations - Gene
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Example 63. − 3x +9=6x − 27 Not
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4[ − 2]+9=−15 +8(2) Finish mult
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1.4 Solving Linear Equations - Frac
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Example 72. � � 3 5 4 x + = 3 2
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1.5 Solving Linear Equations - Form
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Example 79. A = πr 2 +πrs for s S
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1.5 Practice - Formulas Solve each
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|x| = 5 Absolute value can be posit
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Example 90. 7+|2x − 5| =4 Notice
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1.7 Solving Linear Equations - Vari
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Once we have found the constant of
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1.7 Practice - Variation Write the
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sure. If the pressure of a certain
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Fifteen more than three times a num
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When we started with our first, sec
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1.8 Practice - Number and Geometry
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30. The perimeter of a college bask
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Our equation comes from the future
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4x + 2=102 Solve the two − step e
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years ago, the bronze plaque was on
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1.10 Solving Linear Equations - Dis
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Rate Time Distance Bob r + 2 3 Fred
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2 =t Our solution fort, she catches
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Find the distance he rode. 8. A man
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The first plan is flying 25 mph slo
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2.1 Graphing - Points and Lines Obj
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E B D F G C A Our Solution The main
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x y − 3 − 4 0 − 2 3 0 Our com
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2.2 Graphing - Slope Objective: Fin
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When mathematicians began working w
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Example 131. Find the value of x be
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9) 10) Find the slope of the line t
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y = − 2 x + 3 Our Solution 3 Iden
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2.3 Practice - Slope-Intercept Writ
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2.4 Graphing - Point-Slope Form Obj
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Find the equation of the line throu
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Write the point-slope form of the e
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m = 2 5 m = 2 5 Example 147. Slope
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2.5 Practice - Parallel and Perpend
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Chapter 3 : Inequalities 3.1 Solve
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than 4. If we have an expression su
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− 5 − 5 − 2x � 6 Divide bot
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Solve each inequality, graph each s
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As the graphs overlap, we take the
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3.2 Practice - Compound Inequalitie
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edition of a college algebra text!
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It is important to remember as we a
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Chapter 4 : Systems of Equations 4.
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ested in the point that is a soluti
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that parallel lines have the same s
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4.2 Systems of Equations - Substitu
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tion. This process is described and
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World View Note: French mathematici
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31) 2x + y = 2 3x + 7y = 14 33) x +
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6x +9y = 6 New second equation −
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Just as with graphing and substutio
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4.4 Systems of Equations - Three Va
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just the ordered pair (x, y). In th
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15x − 12y +9z = − 12 5x − 4y
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23) 3x + 3y − 2z = 13 6x + 2y −
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are given to use. This means someti
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− 0.5 − 0.5 c=17 We have c, num
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− 0.06x − 0.06y = − 240 Add e
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14) A bank contains 27 coins in dim
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4.6 Systems of Equations - Mixture
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Amount Part Total Start 40 3 120 Ad
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4(30 −n)+2.5n=105 Substitute into
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16) A certain grade of milk contain
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175
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5.1 Polynomials - Exponent Properti
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A quicker method to arrive at the s
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Example 210. Example 211. 7a 3 (2a
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5.2 Polynomials - Negative Exponent
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As we simplified our fraction we to
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5.2 Practice - Negative Exponents S
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Example 223. Convert 3.21 × 10 5 t
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5.3 Practice - Scientific Notation
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World View Note: Ada Lovelace in 18
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21) (3+b 4 )+(7+2b + b 4 ) 22) (1+6
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Just as we distribute a monomial th
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Example 243. (2x − 5)(4x 2 − 7x
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5.6 Polynomials - Multiply Special
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Be very careful when we are squarin
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5.7 Polynomials - Divide Polynomial
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3. Change the sign of the terms and
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Example 261. 2x 3 + 42 − 4x x + 3
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Chapter 6 : Factoring 6.1 Greatest
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ers using mental math, then we take
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6.1 Practice - Greatest Common Fact
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Example 273. 10ab + 15b + 4a+6 Spli
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example the binomials are (a + b) a
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6.3 Factoring - Trinomials where a
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As the past few examples illustrate
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6.3 Practice - Trinomials where a =
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Example 296. 10x 2 − 27x +5 Multi
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6.5 Factoring - Factoring Special P
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Example 306. 4x 2 + 20xy + 25y 2 Mu
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6.5 Practice - Factoring Special Pr
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Example 313. Example 314. Example 3
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6.7 Factoring - Solve by Factoring
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0=(2x +3)(2x − 3) One factor must
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6.7 Practice - Solve by Factoring S
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7.1 Rational Expressions - Reduce R
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However, if there is more than just
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41) 8m+16 20m − 12 43) 2x2 − 10
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Example 333. 25x2 24y4 · 9y8 55x7
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Simplify each expression. 1) 8x2 9
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7.3 Rational Expressions - Least Co
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As the above example illustrates, w
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7.4 Rational Expressions - Add & Su
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13 12 Our Solution The same process
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7.4 Practice - Add and Subtract Add
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2(12) 1(12) − 3 4 5(12) 1(12) + 6
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LCD =(x + 3)(x − 3) Multiply each
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23) 25) 27) 29) x − 4+ 9 2x +3 x
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(20)(9)=6x Multiply 180 = 6x Divide
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3.5 6 The man hasashadow of 3.5 fee
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33) Kali reduced the size of a pain
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LCD will be more involved. We will
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x=5 or − 2 Our Solution World Vie
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7.8 Rational Expressions - Dimensio
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� 435 1 �� 1 lbs = 16 435
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To focus on the process of conversi
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7.8 Practice - Dimensional Analysis
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Chapter 8 : Radicals 8.1 Square Roo
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√ process is being able to transl
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Simplify. √ 1) 245 √ 3) 36 √
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Example 384. 3√ 54 We are working
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8.3 Radicals - Adding Radicals Obje
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Simiplify 1) 2 5 √ + 2 5 √ + 2
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√ √ √ √ 4 50 + 6 30 − 8 3
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The previous example could have bee
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8.5 Radicals - Rationalize Denomina
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2x2 √ √ 10 − 6x 3x Our Soluti
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Example 408. 2 5 √ − 3 7 √ 5
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37) 5 2 √ + 3 √ 5+5 2 √ 38) 3
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1 (3) 4 Evaluate exponent 1 81 Our
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8.6 Practice - Rational Exponents W
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Example 420. ab2 3√ a2 4√ b Rew
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8.7 Practice - Radicals of Mixed In
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Example 426. Example 427. i 35 Divi
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√ Dividing with complex numbers a
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Simplify. 1) 3 − ( − 8+4i) 3) (
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Chapter 9 : Quadratics 9.1 Solving
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Example 444. + 1 +1 Add1to both sid
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2 − 2=0 Subtract 0=0 True! It wor
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Solve. √ 1) 2x + 3 √ 3) 6x −
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x4 4√ = ± 4√ 16 Simplify roots
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5√ 4x +1 = ± 3 Clear radical by
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9.3 Quadratics - Complete the Squar
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x2 + 5 25 x+ 3 36 � x + 5 �2 6
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Example 464. − 7 3 � 3 3 + 3 3
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9.4 Quadratics - Quadratic Formula
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x = Example 468. √ 30 ± 900 + 11
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9.4 Practice - Quadratic Formula So
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12x 2 − 17x + 6=0 Our Solution If
Page 351 and 352: 9.5 Practice - Build Quadratics fro
Page 353 and 354: Example 479. a −2 − a −1 −
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Page 357 and 358: 9.7 Quadratics - Rectangles Objecti
Page 359 and 360: Example 486. 2 Our Solution The len
Page 361 and 362: 4(x 2 + 10x − 24) =0 Factor trino
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Page 365 and 366: 1(12x) 3 + 1(12x) x 1 1 5 + = 3 x 1
Page 367 and 368: 6x +6(x − 1)=5x(x − 1) Reduce e
Page 369 and 370: 16) A sink can be filled from the f
Page 371 and 372: These simultaneous product equation
Page 373 and 374: 9.10 Quadratics - Revenue and Dista
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Page 379 and 380: 12) A pilot flying at a constant ra
Page 381 and 382: The above method to graph a parabol
Page 383 and 384: It is important to remember the gra
Page 385 and 386: Chapter 10 : Functions 10.1 Functio
Page 387 and 388: Example 504. Which of the following
Page 389 and 390: 2x − 3 � 0 Set up an inequality
Page 391 and 392: Solve. 10.1 Practice - Function Not
Page 393 and 394: 10.2 Functions - Operations on Func
Page 395 and 396: (x − 5)(x +1) (x − 5) Divide ou
Page 397 and 398: We can also evaluate a composition
Page 399 and 400: 23) f(x) = x 2 − 5x g(x) = x + 5
Page 401: 10.3 Functions - Inverse Functions
Page 405 and 406: 10.3 Practice - Inverse Functions S
Page 407 and 408: Example 530. 8 3x = 32 Rewrite 8 as
Page 409 and 410: 10.4 Practice - Exponential Functio
Page 411 and 412: log3x =7 Identify base, 3, answer,
Page 413 and 414: 10.5 Practice - Logarithmic Functio
Page 415 and 416: A = 25000(1.01625) 20 Evaluate expo
Page 417 and 418: The variable e is a constant simila
Page 419 and 420: j. All of the above compounded cont
Page 421 and 422: Using the diagram at right, find ea
Page 423 and 424: 10.7 Practice - Trigonometric Funct
Page 425 and 426: 21) 23) 25) 27) A B A 5 38 ◦ x x
Page 427 and 428: 37) 39) A A 37.1 ◦ C x x C 13.1 4
Page 429 and 430: θ 17 From angle θ the given sides
Page 431 and 432: function to find the missing angle
Page 433 and 434: Find the measure of each angle indi
Page 435 and 436: 27) 29) 14 A A θ C 15 7 C 14 θ B
Page 437 and 438: 437
Page 439 and 440: 7) 5 4 8) 4 3 9) 3 2 10) 8 3 11) 5
Page 441 and 442: 54) 60v − 7 55) − 3x + 8x 2 56)
Page 443 and 444: 45) 12 46) All real numbers 47) No
Page 445 and 446: 15) wx 3 = 1458 16) h = 1.5 j 17) a
Page 447 and 448: 1) B(4, − 3) C(1, 2) D( − 1, 4)
Page 449 and 450: 1) y = 2x +5 2) y = − 6x +4 3) y
Page 451 and 452: 23) x =4 24) y − 4 = 7 (x − 1)
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25) 5�x
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1) ( − 2, 4) 2) (2,4) 3) No solut
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27) 56, 144 28) 1.5, 3.5 29) 30 30)
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42) 1.2 × 10 6 5.4 1) 3 2) 7 3)
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5) 2x 3 + 4x 2 + x 2 6) 5p3 4 +4p2
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7) (b +8)(b + 4) 8) (b − 10)(b
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11) n(n − 1) 12) (5x +3)(x − 5)
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5) 3x2 2 6) 5p 2 7) 5m 8) 7 10 9) r
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7) 8) 5 24r 7x + 3y x 2 y 2 9) 15t+
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21) 0, 5 22) − 2, 5 3 23) 4,7 24)
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42) − 18xz 4x3yz3 � 4 8.3 1) 6
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16 3 √ + 4 5 √ 43 18) 19) 2 √
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8.8 1) 11 − 4i 2) − 4i 3) − 3
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√ √ 33) 4+i 39, 4 −i 39 34)
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9) ± 2, ± 4 10) 2,3, − 1 ±i 3
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1) 2) 3) 4) 5) 6) (-2,0) (0,-8) (-1
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11) 100 12) − 74 13) 1 5 14) 27 1
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10.5 1) 9 2 = 81 2) b −16 = a 3)
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21) 51.3 ◦ 22) 45 ◦ 23) 56.4
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