Problem

If $ f(x) = x + 4 $ and $ h(x) = 4x - 1 $, find a…

01:14

Question

Answered step-by-step

Problem 63 Hard Difficulty

(a) If $ g(x) = 2x + 1 $ and $ h(x) = 4x^2 + 4x + 7 $, find a function $ f $ such that $ f \circ g = h $. (Think about what operations you would have to perform on the formula for $ g $ to end up with the formula for $ h $.)
(b) If $ f(x) = 3x + 5 $ and $ h(x) = 3x^2 + 3x + 2 $, find a function $ g $ such that $ f \circ g = h $.

Video Answer

Solved by verified expert

Textbook Answer

Official textbook answer

Video by Heather Zimmers

Numerade Educator

This textbook answer is only visible when subscribed! Please subscribe to view the answer

Johns Hopkins University

Watch More Solved Questions in Chapter 1

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Problem 26

Problem 27

Problem 28

Problem 29

Problem 30

Problem 31

Problem 32

Problem 33

Problem 34

Problem 35

Problem 36

Problem 37

Problem 38

Problem 39

Problem 40

Problem 41

Problem 42

Problem 43

Problem 44

Problem 45

Problem 46

Problem 47

Problem 48

Problem 49

Problem 50

Problem 51

Problem 52

Problem 53

Problem 54

Problem 55

Problem 56

Problem 57

Problem 58

Problem 59

Problem 60

Problem 61

Problem 62

Problem 63

Problem 64

Problem 65

Problem 66

Video Transcript

So here we have function h of X, and we know that h of X is f f G G is the inside function and we know it. But we don't know the outside function. That's what we're going to find. We're going to find F. So let's find a way to rewrite H. And when you see four x squared here and you see two X here, hopefully you realize that in some way we must have squared G. So let's figure out what G squared is. G squared would be two x plus one quantity squared, and that would be four x squared plus four x plus one. So let's think of age as four X squared plus four x plus one plus six. We have to get that seven in there, so we need one in six to give us seven. And now we can rewrite that perfect try no meal square as two x plus one quantity squared and then we have the plus six. So if we think of age in this way, we can see the inside function. G is two x plus one on the outside function. Must be. Then what? We get when we square that and then add six. So f of X is X squared plus six for part B. We have a similar situation where we have a function h of X, and now we know the outside function f but we don't know the inside function G. That's what we have to figure out. So what we want to do is work on H and get it to look more like this three times something plus five. So if we change h a little bit by factoring out of three, we have three times X squared plus X plus two. Well, we need that to be plus five on the end, not plus two. So we need to add another three. But how can we just add three? We can't just add three unless we also subtract three. So what I'm going to dio is incorporate this minus three into the inside of the parentheses in some way. So if I write it as three times a quantity X squared plus X minus one and then the two plus three on the outside is going to be plus five. Here we have our minus three on the inside compensating for are extra plus three on the outside. So we have three times a quantity X squared plus X minus one plus five that is H of X, rewritten in a different form. Now that matches with our f of X three times something plus five. So that's something on the inside that must be RG so g of X is X squared plus X minus one.

Get More Help with this Textbook

James Stewart

Calculus: Early Transcendentals

View More Answers From This Book

Find Another Textbook

Study Groups

Study with other students and unlock Numerade solutions for free.

Problem

If $ f(x) = x + 4 $ and $ h(x) = 4x - 1 $, find a…

This textbook answer is only visible when subscribed! Please subscribe to view the answer

More Answers

Related Courses

Calculus: Early Transcendentals

Related Topics

Discussion

Top Calculus 3 Educators

Lily An

Anna Marie Vagnozzi

Samuel Hannah

Michael Jacobsen

Calculus 3 Courses

Recommended Videos

Watch More Solved Questions in Chapter 1

Video Transcript

James Stewart

Calculus: Early Transcendentals

Lily An

Anna Marie Vagnozzi

Samuel Hannah

Michael Jacobsen

This textbook answer is only visible when subscribed! Please subscribe to view the answer

Calculus: Early Transcendentals

Lily An

Anna Marie Vagnozzi

Samuel Hannah

Michael Jacobsen

James StewartCalculus: Early Transcendentals

Lily An

Anna Marie Vagnozzi

Samuel Hannah

Michael Jacobsen

James Stewart

Calculus: Early Transcendentals