Advanced Functions 12
Advanced Functions 12
1st Edition
Chris Kirkpatrick, Kristina Farentino, Susanne Trew
ISBN: 9780176678326
Table of contents
Textbook solutions

All Solutions

Section 9-3: Combining Two Functions: Products

Exercise 1
Step 1
1 of 2
#### (a)

$(ftimes g)(x)={(0,2 times -1), (1,5 times -2), (2,7 times 3), (3,12 times 5)}$={(0, -2), (1, -10), (2, 21), (3,60)}

#### (b)

$(ftimes g)(x)={(0,3 times 4), (2,10 times -2)}={(0,12), (2, -20)}$

$(ftimes g)(x)=x(4)=4x$

#### (d)

$(ftimes g)(x)=x(2x)=2x^2$

#### (e)

$(ftimes g)(x)=(x+2)(x^2-2x+1)$

$=x^3-2x^2+x+2x^2-4x+2$

$=x^3-3x+2$

#### (f)

$(ftimes g)(x)=2^x(sqrt{x-2})$

$=2^xsqrt{x-2}$

Result
2 of 2
see solution
Exercise 2
Step 1
1 of 9
#### (a)

1.task, part (c), red is graph of the function $f(x)$ and blue is of $g(x)$.

Exercise scan

Step 2
2 of 9
1.task, part (d), red is graph of the function $f(x)$ and blue is of $g(x)$.

Exercise scan

Step 3
3 of 9
1.task, part (e), red is graph of the function $f(x)$ and blue is of $g(x)$.

Exercise scan

Step 4
4 of 9
1.task, part (f), red is graph of the function $f(x)$ and blue is of $g(x)$.

Exercise scan

Step 5
5 of 9
#### (b)

1.task, part (c), $f:left{xinBbb{R} right}$, $g:left{xinBbb{R} right}$

1.task, part (d), $f:left{xinBbb{R} right}$, $g:left{xinBbb{R} right}$

1.task, part (e), $f:left{xinBbb{R} right}$, $g:left{xinBbb{R} right}$

1.task, part (f), $f:left{xinBbb{R} right}$, $g:left{xinBbb{R}|xgeq2 right}$

#### (c)

The graph of $ftimes{g}$ can be found by multipliying corresponding $y-coordinates$.

1.task, part (c), on the following picture there is a graph of $ftimes{g}$:

Exercise scan

Step 6
6 of 9
1.task, part (d), on the following picture there is a graph of $ftimes{g}$:

Exercise scan

Step 7
7 of 9
1.task, part (e), on the following picture there is a graph of $ftimes{g}$:

Exercise scan

Step 8
8 of 9
1.task, part (f), on the following picture there is a graph of $ftimes{g}$:

Exercise scan

Result
9 of 9
see solution
Exercise 3
Step 1
1 of 2
$(ftimes g )(x)=(sqrt{1+x})(sqrt{1-x})$

So,$xgeq-1$ and $xleq 1$ since the radicand must be greater than or equal to $0$.

So,the domain is $left{xinBbb{R}|-1leq x leq 1 right}$

Result
2 of 2
see solution
Exercise 4
Step 1
1 of 2
#### (a)

$(ftimes g)(x)=(x-7)(x+7)=x^2-49$

#### (b)

$(ftimes g)(x)=(sqrt{x+10})(sqrt{x+10})=x+10$

#### (c)

$(ftimes g)(x)=7x^2(x-9)=7x^3-63x^2$

#### (d)

$(ftimes g)(x)=(-4x-7)(4x+7)=-16x^2-28x-28x-49=16x^2-56x-49$

#### (e)

$(ftimes g)(x)=2sin xleft(dfrac{1}{x-1} right)=dfrac{2sin x}{x-1}$

#### (f)

$(ftimes g)(x)=log (x+4)(2^x)=2^x log (x+4)$

Result
2 of 2
see solution
Exercise 5
Step 1
1 of 2
4(a): $D=left{xinBbb{R} right}$; $R=left{y in Bbb{R}|ygeq-49 right}$

4(b): $D=left{xinBbb{R}|xgeq -10 right}$; $R=left{yin Bbb{R}|ygeq0 right}$

4(c): $D=left{ xinBbb{R}right}$; $R=left{yinBbb{R} right}$

4(d): $D=left{ xinBbb{R}right}$; $R=left{yinBbb{R}|yleq0 right}$

4(e): $D=left{ xinBbb{R} | xne -1right}$; $R=left{yin Bbb{R} right}$

4(f): $D=left{xinBbb{R}|x > -4 right}$; $R=left{yinBbb{R}|ygeq 0 right}$

Result
2 of 2
see solution
Exercise 6
Step 1
1 of 2
4(a): symmetry: The function is symmetric about the line $x=0$.
increasing/decreasing: The function is increasing from $0$to $infty$. The function is is decreasing from $-infty$ to $0$.
zeros: $x=-7, 7$

maximum/minimum: The minimum is at $x=0$.

period: N/A

4(b): symmetry: The function is not symmetric.
increasing/decreasing: The function is increasing from $-10$ to $infty$.

zeros: $x= -10$

maximum/minimum: The minimum is at $x=-10$.

period: N/A

4(c): symmetry: The function is not symmetic.
increasing/decreasing: The function is increasing from $-infty$to $0$ and from $-infty$ to $0$ and from $6$ to $infty$.

zeros: $x=0,9$

maximum/minimum: The relative minimum is at $x=-6$. The relative maximum is at $x=0$.

period: N/A

4(d): symmetry: The function is symmetric about the line $x=-1.75$.

increasing/decreasing: The function is increasing from $-infty$ to $-1.75$ and is decreasing from $-1.75$ to $infty$.

zero: $x=-1.75$

maximum/minimum: The maximum is at $x=-1.75$.

period: N/A

4(e): symmetry: The function is not symmetric.
increasing/decreasing: The function is increasing from $-infty$ to $0$ and from $6$ to $infty$.

zeros: $x=0,9$

maximum/minimum: The relative minima are at $x=-4.5336$ and $4.4286$.The relative maximum is at $x=-1.1323$.

period: N/A

4(f): symmetry: The function is not symmetric.
increasing/decreasing: The function is increasing from $-4$ to $infty$.

zeros: none

maximum/minimum: none

period: N/A

Result
2 of 2
see solution
Exercise 7
Step 1
1 of 2
$f(x)=-4x$

$g(x)=6x+1$

$(ftimes g)(x)= -4x(6x+1)=-24x^2-4x$

Exercise scan

Result
2 of 2
see solution
Exercise 8
Step 1
1 of 2
#### (a)

$left{xinBbb{R}|xne -2,7, dfrac{pi}{2},or dfrac{3pi}{2} right}$

#### (b)

$left{xinBbb{R}|x>8 right}$

#### (c)

$left{xinBbb{R}|xgeq-81 (and) xne0,pi, or 2pi right}$

#### (d)

$$
left{xinBbb{R}|xleq -1 or x geq 1 (and) xne -3 right}
$$

Result
2 of 2
see solution
Exercise 9
Step 1
1 of 2
$(ftimes p)(t)$ represents the total energy consumption in a particular country at time $t$.
Result
2 of 2
$(ftimes p)(t)$
Exercise 10
Step 1
1 of 2
#### (a)

$R(x)=(20 000-750x)(25+x)$ or $R(x)=500 000+1250x-750x^2$, where $x$ is the increase in the admission fee in dollars

#### (b)

Yes, it’s the product of the function $P(x)=20000-750x$, which represents the number of daily visitors, and $F(x)=25+x$, which represents the admission fee.

#### (c)

Use a graphing calculator. The ticket price that will maximize revenue is 25.83$.

Result
2 of 2
see solution
Exercise 11
Step 1
1 of 2
$m(t)=((0.9)^t)(650+300t)$

Use a graphing calculator to estimate.

$$
textbf{The amount of contaminated material is at its greatest after about $7.3$s.}
$$

Result
2 of 2
$7.3$s
Exercise 12
Step 1
1 of 2
The statement is false. If $f(x)$ and $g(x)$ are odd functions, then their product will always be an even function. The reason is because when you multiply a function that has an odd degree with another function that has an odd degree, you add the exponents and when you add two odd numbers together, you get an even number.
Result
2 of 2
see solution
Exercise 13
Step 1
1 of 2
$h(x)=(mx^2+2x+5)(2x^2-nx-2)$

$-40=(m(1)^2+2(1)+5) times(2(1)^2-n(1)-2)$

$-40=(m+2+5)(2-n-2)$

$-40=(m+7)(-n)$

$dfrac{40}{m+7}=n$ EQUATION 1

$24=(m(-1)^2+2(-1)+5)times (2(-1)^2-n(-1)-2)$

$24=(m-2+5)(2+n-2)$

$24=(m+3)(n)$

$dfrac{24}{m+3}=n$ EQUATION 2

$dfrac{40}{m+7}=dfrac{24}{m+3}$

$40(m+3)=24(m+7)$

$40m+120=24m+168$

$16m=48$

$m=3$

$dfrac{24}{3+3}=n$

$dfrac{24}{6}=n$

$4=n$

$$
textbf{So, the equations are $f(x)=3x^2+2x+5$ and $g(x)=2x^2-4x-2$.}
$$

Result
2 of 2
$f(x)=3x^2+2x+5$ and $g(x)=2x^2-4x-2$
Exercise 14
Step 1
1 of 2
#### (a)

$(ftimes g)(x)=sqrt{-x}log(x+10)$

The domain is $left{xinBbb{R}| -10 < x leq 0 right}$.

#### (b)

One strategy is to create a table of values for $f(x)$ and $g(x)$ and to multiply the corresponding $y$-values together. The resulting values could then be graphed.Another strategy is to graph $f(x)$ and $g(x)$ and to then create a graph for $(ftimes g)(x)$ based on these two graphs. The first strategy is probably better than the second strategy, since the $y$-values for $f(x)$ and $g(x)$ will not be round numbers and will not be easily discernable from the graphs of $f(x)$ and $g(x)$.

#### (c)

Exercise scan

Result
2 of 2
see solution
Exercise 15
Step 1
1 of 2
#### (a)

$f(x)times dfrac{1}{f(x)}=(x^2-25)times dfrac{1}{x^2-25}$

#### (b)

The domain of the function is $left{xinBbb{R}|xne -5 (or) 5 right}$.

#### (c)

The range will always be $1$. If $f$ is of odd degree, there will always be at least one value that makes the product undefined and which is excluded from the domain. If $f$ is of even degree, there may be no values that are excluded from domain.

Exercise scan

Result
2 of 2
see solution
Exercise 16
Step 1
1 of 2
#### (a)

$f(x)=2^x$

$g(x)=x^2+1$

$(ftimes g)(x)=2^x(x^2+1)$

#### (b)

$f(x)=x$

$g(x)=sin(2pi x)$

$(ftimes g)(x)=x sin(2pi x)$

Result
2 of 2
see solution
Exercise 17
Step 1
1 of 3
#### (a)

$4x^2-91=(2x+9)(2x-9)$

$f(x)=(2x+9)$

$g(x)=(2x-9)$

#### (b)

$8sin^3 x+27=(2sin x +3)times(4sin^2x-6sin x+9)$

$f(x)=(2sin x+3)$

$g(x)=(4sin^2x-6sin x+9)$

Step 2
2 of 3
#### (c)

$4x^{dfrac{5}{2}}-3x^{dfrac{3}{2}}+x^{dfrac{1}{2}}=x^{dfrac{1}{2}}(4x^5-3x^3+1)$

$f(x)=x^{dfrac{1}{2}}$

$g(x)=(4x^5-3x^3+1)$

#### (d)

$dfrac{6x-5}{2x+1}=dfrac{1}{2x+1}times 6x-5$

$f(x)=dfrac{1}{2x+1}$

$g(x)=6x-5$

Result
3 of 3
see solution
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Chapter 1: Functions: Characteristics and Properties
Page 2: Getting Started
Section 1-1: Functions
Section 1-2: Exploring Absolute Value
Section 1-3: Properties of Graphs of Functions
Section 1-4: Sketching Graphs of Functions
Section 1-5: Inverse Relations
Section 1-6: Piecewise Functions
Section 1-7: Exploring Operations with Functions
Page 62: Chapter Self-Test
Chapter 2: Functions: Understanding Rates of Change
Page 66: Getting Started
Section 2-1: Determining Average Rate of Change
Section 2-2: Estimating Instantaneous Rates of Change from Tables of Values and Equations
Section 2-3: Exploring Instantaneous Rates of Change Using Graphs
Section 2-4: Using Rates of Change to Create a Graphical Model
Section 2-5: Solving Problems Involving Rates of Change
Page 118: Chapter Self-Test
Chapter 3: Polynomial Functions
Page 122: Getting Started
Section 3-1: Exploring Polynomial Functions
Section 3-2: Characteristics of Polynomial Functions
Section 3-3: Characteristics of Polynomial Functions in Factored Form
Section 3-4: Transformation of Cubic and Quartic Functions
Section 3-5: Dividing Polynomials
Section 3-6: Factoring Polynomials
Section 3-7: Factoring a Sum or Difference of Cubes
Page 186: Chapter Self-Test
Page 188: Cumulative Review
Page 155: Check Your Understanding
Page 161: Practice Questions
Page 182: Check Your Understanding
Page 184: Practice Questions
Chapter 4: Polynomial Equations and Inequalities
Page 194: Getting Started
Section 4-1: Solving Polynomial Equations
Section 4-2: Solving Linear Inequalities
Section 4-3: Solving Polynomial Inequalities
Section 4-4: Rates of Change in Polynomial Functions
Page 242: Chapter Self-Test
Chapter 5: Rational Functions, Equations, and Inequalities
Page 246: Getting Started
Section 5-1: Graphs of Reciprocal Functions
Section 5-2: Exploring Quotients of Polynomial Functions
Section 5-3: Graphs of Rational Functions of the Form f(x) 5 ax 1 b cx 1 d
Section 5-4: Solving Rational Equations
Section 5-5: Solving Rational Inequalities
Section 5-6: Rates of Change in Rational Functions
Page 310: Chapter Self-Test
Chapter 6: Trigonometric Functions
Page 314: Getting Started
Section 6-1: Radian Measure
Section 6-2: Radian Measure and Angles on the Cartesian Plane
Section 6-3: Exploring Graphs of the Primary Trigonometric Functions
Section 6-4: Transformations of Trigonometric Functions
Section 6-5: Exploring Graphs of the Reciprocal Trigonometric Functions
Section 6-6: Modelling with Trigonometric Functions
Section 6-7: Rates of Change in Trigonometric Functions
Page 378: Chapter Self-Test
Page 380: Cumulative Review
Chapter 7: Trigonometric Identities and Equations
Page 386: Getting Started
Section 7-1: Exploring Equivalent Trigonometric Functions
Section 7-2: Compound Angle Formulas
Section 7-3: Double Angle Formulas
Section 7-4: Proving Trigonometric Identities
Section 7-5: Solving Linear Trigonometric Equations
Section 7-6: Solving Quadratic Trigonometric Equations
Page 441: Chapter Self-Test
Chapter 8: Exponential and Logarithmic Functions
Page 446: Getting Started
Section 8-1: Exploring the Logarithmic Function
Section 8-2: Transformations of Logarithmic Functions
Section 8-3: Evaluating Logarithms
Section 8-4: Laws of Logarithms
Section 8-5: Solving Exponential Equations
Section 8-6: Solving Logarithmic Equations
Section 8-7: Solving Problems with Exponential and Logarithmic Functions
Section 8-8: Rates of Change in Exponential and Logarithmic Functions
Page 512: Chapter Self-Test
Chapter 9: Combinations of Functions
Page 516: Getting Started
Section 9-1: Exploring Combinations of Functions
Section 9-2: Combining Two Functions: Sums and Differences
Section 9-3: Combining Two Functions: Products
Section 9-4: Exploring Quotients of Functions
Section 9-5: Composition of Functions
Section 9-6: Techniques for Solving Equations and Inequalities
Section 9-7: Modelling with Functions
Page 578: Chapter Self-Test
Page 580: Cumulative Review
Page 542: Further Your Understanding
Page 544: Practice Questions
Page 569: Check Your Understanding
Page 576: Practice Questions