# 1.4 Composition of functions  (Page 8/9)

 Page 8 / 9

$h\left(x\right)={\left(x+2\right)}^{2}$

$h\left(x\right)={\left(x-5\right)}^{3}$

sample: $\begin{array}{l}f\left(x\right)={x}^{3}\\ g\left(x\right)=x-5\end{array}$

$h\left(x\right)=\frac{3}{x-5}$

$h\left(x\right)=\frac{4}{{\left(x+2\right)}^{2}}$

sample: $\begin{array}{l}f\left(x\right)=\frac{4}{x}\hfill \\ g\left(x\right)={\left(x+2\right)}^{2}\hfill \end{array}$

$h\left(x\right)=4+\sqrt[3]{x}$

$h\left(x\right)=\sqrt[3]{\frac{1}{2x-3}}$

sample: $\begin{array}{l}f\left(x\right)=\sqrt[3]{x}\\ g\left(x\right)=\frac{1}{2x-3}\end{array}$

$h\left(x\right)=\frac{1}{{\left(3{x}^{2}-4\right)}^{-3}}$

$h\left(x\right)=\sqrt[4]{\frac{3x-2}{x+5}}$

sample: $\begin{array}{l}f\left(x\right)=\sqrt[4]{x}\\ g\left(x\right)=\frac{3x-2}{x+5}\end{array}$

$h\left(x\right)={\left(\frac{8+{x}^{3}}{8-{x}^{3}}\right)}^{4}$

$h\left(x\right)=\sqrt{2x+6}$

sample: $f\left(x\right)=\sqrt{x}$
$g\left(x\right)=2x+6$

$h\left(x\right)={\left(5x-1\right)}^{3}$

$h\left(x\right)=\sqrt[3]{x-1}$

sample: $f\left(x\right)=\sqrt[3]{x}$
$g\left(x\right)=\left(x-1\right)$

$h\left(x\right)=|{x}^{2}+7|$

$h\left(x\right)=\frac{1}{{\left(x-2\right)}^{3}}$

sample: $f\left(x\right)={x}^{3}$
$g\left(x\right)=\frac{1}{x-2}$

$h\left(x\right)={\left(\frac{1}{2x-3}\right)}^{2}$

$h\left(x\right)=\sqrt{\frac{2x-1}{3x+4}}$

sample: $f\left(x\right)=\sqrt{x}$
$g\left(x\right)=\frac{2x-1}{3x+4}$

## Graphical

For the following exercises, use the graphs of $\text{\hspace{0.17em}}f,$ shown in [link] , and $\text{\hspace{0.17em}}g,$ shown in [link] , to evaluate the expressions.

$f\left(g\left(3\right)\right)$

$f\left(g\left(1\right)\right)$

2

$g\left(f\left(1\right)\right)$

$g\left(f\left(0\right)\right)$

5

$f\left(f\left(5\right)\right)$

$f\left(f\left(4\right)\right)$

4

$g\left(g\left(2\right)\right)$

$g\left(g\left(0\right)\right)$

0

For the following exercises, use graphs of $\text{\hspace{0.17em}}f\left(x\right),$ shown in [link] , $\text{\hspace{0.17em}}g\left(x\right),$ shown in [link] , and $\text{\hspace{0.17em}}h\left(x\right),$ shown in [link] , to evaluate the expressions.

$g\left(f\left(1\right)\right)$

$g\left(f\left(2\right)\right)$

2

$f\left(g\left(4\right)\right)$

$f\left(g\left(1\right)\right)$

1

$f\left(h\left(2\right)\right)$

$h\left(f\left(2\right)\right)$

4

$f\left(g\left(h\left(4\right)\right)\right)$

$f\left(g\left(f\left(-2\right)\right)\right)$

4

## Numeric

For the following exercises, use the function values for shown in [link] to evaluate each expression.

 $x$ $f\left(x\right)$ $g\left(x\right)$ 0 7 9 1 6 5 2 5 6 3 8 2 4 4 1 5 0 8 6 2 7 7 1 3 8 9 4 9 3 0

$f\left(g\left(8\right)\right)$

$f\left(g\left(5\right)\right)$

9

$g\left(f\left(5\right)\right)$

$g\left(f\left(3\right)\right)$

4

$f\left(f\left(4\right)\right)$

$f\left(f\left(1\right)\right)$

2

$g\left(g\left(2\right)\right)$

$g\left(g\left(6\right)\right)$

3

For the following exercises, use the function values for shown in [link] to evaluate the expressions.

 $x$ $f\left(x\right)$ $g\left(x\right)$ -3 11 -8 -2 9 -3 -1 7 0 0 5 1 1 3 0 2 1 -3 3 -1 -8

$\left(f\circ g\right)\left(1\right)$

$\left(f\circ g\right)\left(2\right)$

11

$\left(g\circ f\right)\left(2\right)$

$\left(g\circ f\right)\left(3\right)$

0

$\left(g\circ g\right)\left(1\right)$

$\left(f\circ f\right)\left(3\right)$

7

For the following exercises, use each pair of functions to find $\text{\hspace{0.17em}}f\left(g\left(0\right)\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(f\left(0\right)\right).$

$f\left(x\right)=4x+8,\text{\hspace{0.17em}}g\left(x\right)=7-{x}^{2}$

$f\left(x\right)=5x+7,\text{\hspace{0.17em}}g\left(x\right)=4-2{x}^{2}$

$f\left(g\left(0\right)\right)=27,\text{\hspace{0.17em}}g\left(f\left(0\right)\right)=-94$

$f\left(x\right)=\sqrt{x+4},\text{\hspace{0.17em}}g\left(x\right)=12-{x}^{3}$

$f\left(x\right)=\frac{1}{x+2},\text{\hspace{0.17em}}g\left(x\right)=4x+3$

$f\left(g\left(0\right)\right)=\frac{1}{5},\text{\hspace{0.17em}}g\left(f\left(0\right)\right)=5$

For the following exercises, use the functions $\text{\hspace{0.17em}}f\left(x\right)=2{x}^{2}+1\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)=3x+5\text{\hspace{0.17em}}$ to evaluate or find the composite function as indicated.

$f\left(g\left(2\right)\right)$

$f\left(g\left(x\right)\right)$

$18{x}^{2}+60x+51$

$g\left(f\left(-3\right)\right)$

$\left(g\circ g\right)\left(x\right)$

$g\circ g\left(x\right)=9x+20$

## Extensions

For the following exercises, use $\text{\hspace{0.17em}}f\left(x\right)={x}^{3}+1\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)=\sqrt[3]{x-1}.$

Find $\text{\hspace{0.17em}}\left(f\circ g\right)\left(x\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(g\circ f\right)\left(x\right).\text{\hspace{0.17em}}$ Compare the two answers.

Find $\text{\hspace{0.17em}}\left(f\circ g\right)\left(2\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(g\circ f\right)\left(2\right).$

2

What is the domain of $\text{\hspace{0.17em}}\left(g\circ f\right)\left(x\right)?$

What is the domain of $\text{\hspace{0.17em}}\left(f\circ g\right)\left(x\right)?$

$\left(-\infty ,\infty \right)$

Let $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{x}.$

1. Find $\text{\hspace{0.17em}}\left(f\circ f\right)\left(x\right).$
2. Is $\text{\hspace{0.17em}}\left(f\circ f\right)\left(x\right)\text{\hspace{0.17em}}$ for any function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ the same result as the answer to part (a) for any function? Explain.

For the following exercises, let $\text{\hspace{0.17em}}F\left(x\right)={\left(x+1\right)}^{5},\text{\hspace{0.17em}}$ $f\left(x\right)={x}^{5},\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)=x+1.$

True or False: $\text{\hspace{0.17em}}\left(g\circ f\right)\left(x\right)=F\left(x\right).$

False

True or False: $\text{\hspace{0.17em}}\left(f\circ g\right)\left(x\right)=F\left(x\right).$

For the following exercises, find the composition when $\text{\hspace{0.17em}}f\left(x\right)={x}^{2}+2\text{\hspace{0.17em}}$ for all $\text{\hspace{0.17em}}x\ge 0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)=\sqrt{x-2}.$

$\left(f\circ g\right)\left(6\right);\text{\hspace{0.17em}}\left(g\circ f\right)\left(6\right)$

$\left(f\circ g\right)\left(6\right)=6$ ; $\text{\hspace{0.17em}}\left(g\circ f\right)\left(6\right)=6$

$\left(g\circ f\right)\left(a\right);\text{\hspace{0.17em}}\left(f\circ g\right)\left(a\right)$

$\left(f\circ g\right)\left(11\right);\text{\hspace{0.17em}}\left(g\circ f\right)\left(11\right)$

$\left(f\circ g\right)\left(11\right)=11\text{\hspace{0.17em}},\text{\hspace{0.17em}}\left(g\circ f\right)\left(11\right)=11$

## Real-world applications

The function $\text{\hspace{0.17em}}D\left(p\right)\text{\hspace{0.17em}}$ gives the number of items that will be demanded when the price is $\text{\hspace{0.17em}}p.\text{\hspace{0.17em}}$ The production cost $\text{\hspace{0.17em}}C\left(x\right)\text{\hspace{0.17em}}$ is the cost of producing $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ items. To determine the cost of production when the price is $6, you would do which of the following? 1. Evaluate $\text{\hspace{0.17em}}D\left(C\left(6\right)\right).$ 2. Evaluate $\text{\hspace{0.17em}}C\left(D\left(6\right)\right).$ 3. Solve $\text{\hspace{0.17em}}D\left(C\left(x\right)\right)=6.$ 4. Solve $\text{\hspace{0.17em}}C\left(D\left(p\right)\right)=6.$ The function $\text{\hspace{0.17em}}A\left(d\right)\text{\hspace{0.17em}}$ gives the pain level on a scale of 0 to 10 experienced by a patient with $\text{\hspace{0.17em}}d\text{\hspace{0.17em}}$ milligrams of a pain-reducing drug in her system. The milligrams of the drug in the patient’s system after $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ minutes is modeled by $\text{\hspace{0.17em}}m\left(t\right).\text{\hspace{0.17em}}$ Which of the following would you do in order to determine when the patient will be at a pain level of 4? 1. Evaluate $\text{\hspace{0.17em}}A\left(m\left(4\right)\right).$ 2. Evaluate $\text{\hspace{0.17em}}m\left(A\left(4\right)\right).$ 3. Solve $\text{\hspace{0.17em}}A\left(m\left(t\right)\right)=4.$ 4. Solve $\text{\hspace{0.17em}}m\left(A\left(d\right)\right)=4.$ c A store offers customers a 30% discount on the price $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ of selected items. Then, the store takes off an additional 15% at the cash register. Write a price function $\text{\hspace{0.17em}}P\left(x\right)\text{\hspace{0.17em}}$ that computes the final price of the item in terms of the original price $\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ (Hint: Use function composition to find your answer.) A rain drop hitting a lake makes a circular ripple. If the radius, in inches, grows as a function of time in minutes according to $\text{\hspace{0.17em}}r\left(t\right)=25\sqrt{t+2},\text{\hspace{0.17em}}$ find the area of the ripple as a function of time. Find the area of the ripple at $\text{\hspace{0.17em}}t=2.$ $A\left(t\right)=\pi {\left(25\sqrt{t+2}\right)}^{2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}A\left(2\right)=\pi {\left(25\sqrt{4}\right)}^{2}=2500\pi$ square inches A forest fire leaves behind an area of grass burned in an expanding circular pattern. If the radius of the circle of burning grass is increasing with time according to the formula $\text{\hspace{0.17em}}r\left(t\right)=2t+1,\text{\hspace{0.17em}}$ express the area burned as a function of time, $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ (minutes). Use the function you found in the previous exercise to find the total area burned after 5 minutes. $A\left(5\right)=\pi {\left(2\left(5\right)+1\right)}^{2}=121\pi \text{\hspace{0.17em}}$ square units The radius $\text{\hspace{0.17em}}r,\text{\hspace{0.17em}}$ in inches, of a spherical balloon is related to the volume, $\text{\hspace{0.17em}}V,\text{\hspace{0.17em}}$ by $\text{\hspace{0.17em}}r\left(V\right)=\sqrt[3]{\frac{3V}{4\pi }}.\text{\hspace{0.17em}}$ Air is pumped into the balloon, so the volume after $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ seconds is given by $\text{\hspace{0.17em}}V\left(t\right)=10+20t.$ 1. Find the composite function $\text{\hspace{0.17em}}r\left(V\left(t\right)\right).$ 2. Find the exact time when the radius reaches 10 inches. The number of bacteria in a refrigerated food product is given by $N\left(T\right)=23{T}^{2}-56T+1,\text{\hspace{0.17em}}$ $3 where $\text{\hspace{0.17em}}T$ is the temperature of the food. When the food is removed from the refrigerator, the temperature is given by $T\left(t\right)=5t+1.5,$ where $t$ is the time in hours. 1. Find the composite function $\text{\hspace{0.17em}}N\left(T\left(t\right)\right).$ 2. Find the time (round to two decimal places) when the bacteria count reaches 6752. a. $\text{\hspace{0.17em}}N\left(T\left(t\right)\right)=23{\left(5t+1.5\right)}^{2}-56\left(5t+1.5\right)+1;\text{\hspace{0.17em}}$ b. 3.38 hours #### Questions & Answers I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once Carlos Reply How can you tell what type of parent function a graph is ? Mary Reply generally by how the graph looks and understanding what the base parent functions look like and perform on a graph William if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero William y=x will obviously be a straight line with a zero slope William y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis William y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer. Aaron yes, correction on my end, I meant slope of 1 instead of slope of 0 William what is f(x)= Karim Reply I don't understand Joe Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain." Thomas Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-) Thomas GREAT ANSWER THOUGH!!! Darius Thanks. Thomas Â Thomas It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â Thomas Now it shows, go figure? Thomas what is this? unknown Reply i do not understand anything unknown lol...it gets better Darius I've been struggling so much through all of this. my final is in four weeks 😭 Tiffany this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts Darius thank you I have heard of him. I should check him out. Tiffany is there any question in particular? Joe I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously. Tiffany Sure, are you in high school or college? Darius Hi, apologies for the delayed response. I'm in college. Tiffany how to solve polynomial using a calculator Ef Reply So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right? KARMEL Reply The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26 Rima Reply The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer? Rima I done know Joe What kind of answer is that😑? Rima I had just woken up when i got this message Joe Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that Rima i have a question. Abdul how do you find the real and complex roots of a polynomial? Abdul @abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up Nare This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1 Abdul @Nare please let me know if you can solve it. Abdul I have a question juweeriya hello guys I'm new here? will you happy with me mustapha The average annual population increase of a pack of wolves is 25. Brittany Reply how do you find the period of a sine graph Imani Reply Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period Am if not then how would I find it from a graph Imani by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates. Am you could also do it with two consecutive minimum points or x-intercepts Am I will try that thank u Imani Case of Equilateral Hyperbola Jhon Reply ok Zander ok Shella f(x)=4x+2, find f(3) Benetta f(3)=4(3)+2 f(3)=14 lamoussa 14 Vedant pre calc teacher: "Plug in Plug in...smell's good" f(x)=14 Devante 8x=40 Chris Explain why log a x is not defined for a < 0 Baptiste Reply the sum of any two linear polynomial is what Esther Reply divide simplify each answer 3/2÷5/4 Momo Reply divide simplify each answer 25/3÷5/12 Momo how can are find the domain and range of a relations austin Reply the range is twice of the natural number which is the domain Morolake A cell phone company offers two plans for minutes. Plan A:$15 per month and $2 for every 300 texts. Plan B:$25 per month and $0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money? Diddy Reply 6000 Robert more than 6000 Robert For Plan A to reach$27/month to surpass Plan B's $26.50 monthly payment, you'll need 3,000 texts which will cost an additional$10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...
Gilbert
...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...
Gilbert