Answers
g(x) as given has no inverse because there are instances of two x values giving the same value of g(x). For instance,
x = -1 ⇒ g(-1) = 4 (-1 + 3)² - 8 = 8
x = -5 ⇒ g(-5) = 4 (-5 + 3)² - 8 = 8
Only a one-to-one function can have an inverse. g(x) is not one-to-one.
However, if we restrict the domain of g(x), we can find an inverse over that domain. Let [tex]g^{-1}(x)[/tex] be the inverse of g(x). Then by definition of inverse function,
[tex]g\left(g^{-1}(x)\right) = 4 \left(g^{-1}(x) + 3\right)^2 - 8 = x[/tex]
Solve for the inverse:
[tex]4 \left(g^{-1}(x) + 3\right)^2 - 8 = x[/tex]
[tex]4 \left(g^{-1}(x) + 3\right)^2 = x + 8[/tex]
[tex]\left(g^{-1}(x) + 3\right)^2 = \dfrac{x + 8}4[/tex]
[tex]\sqrt{\left(g^{-1}(x) + 3\right)^2} = \sqrt{\dfrac{x + 8}4}[/tex]
[tex]\left| g^{-1}(x) + 3 \right| = \dfrac{\sqrt{x+8}}2[/tex]
Recall the definition of absolute value:
[tex]|x| = \begin{cases}x & \text{if }x\ge0\\-x&\text{if }x<0\end{cases}[/tex]
This means there are two possible solutions for the inverse of g(x) :
• if [tex]g^{-1}(x) + 3 \ge 0[/tex], then
[tex]g^{-1}(x) + 3 = \dfrac{\sqrt{x+8}}2 \implies g^{-1}(x) = -3+\dfrac{\sqrt{x+8}}2[/tex]
• otherwise, if [tex]g^{-1}(x)+3<0[/tex], then
[tex]-\left(g^{-1}(x) + 3\right) = \dfrac{\sqrt{x+8}}2 \implies g^{-1}(x) = -3-\dfrac{\sqrt{x+8}}2[/tex]
Which we choose as the inverse depends on how we restrict the domain of g(x). For example:
Remember that the inverse must satisfy
[tex]g\left(g^{-1}(x)\right) = x[/tex]
In the first case above, [tex]g^{-1}(x) + 3 \ge 0[/tex], or [tex]g^{-1}(x) \ge -3[/tex]. This suggests that we could restrict the domain of g(x) to be [tex]x \ge -3[/tex].
Then as long as [tex]x \ge -3[/tex], the inverse is
[tex]g^{-1}(x) = -3+\dfrac{\sqrt{x+8}}2[/tex]
Related Questions
PLEASE HELP ME WITH THIS PROBLEM
Drew and Stephen shopped together for school supplies. Drew bought 16 pencils and 8 pens for $20.88. Stephen purchased the same items, but he bought 12 pencils and 4 pens for $13.40.
What was the cost of one pen?
Answers
Answer:
Step-by-step explanation:
16x + 8y = 20.88
12x + 4y = 13.40
Solve for x and y
x is the cost of 1 pen
Please help (Math)......
Answers
[tex]|x|^2 -2|x|-3=0\\\\\implies |x|^2 - 3|x| + |x| - 3 =0\\\\\implies |x|(|x| -3) + (|x| -3) = 0\\\\\implies (|x| - 3)(|x| +1) =0\\\\\implies |x| -3 = 0~~ \text{Or} ~~ |x| +1 =0\\\\\implies |x| = 3 ~~ \text{Or}~~ |x| = -1;\\\\\text{For}~ |x|= -1 ,~\text{ there are no solution for } x \in \mathbb{R}\\\\\text{So,}\\\\|x| = 3 \implies x = \pm 3[/tex]
simplify the quotient. Write the answer in sn.
Answers
Answer: 2.25 x 10 -7 -7 is the exponent
Step-by-step explanation:
Move the decimal so there is one non-zero digit to the left of the decimal point. The number of decimal places you move will be the exponent on the
10 If the decimal is being moved to the right, the exponent will be negative. If the decimal is being moved to the left, the exponent will be positive.
LOOKING FOR THE ANSWER TO # 6Saving money is something that everyone should do, if they are able, even if the amounts are small. In
this activity, you will calculate savings based on patterns and graph and compare different functions that
represent savings.
Below is a savings plan for Jeanette. Use this table to answer the questions below.
# of Months Saving Account Balance
2 $70
4 $90
6 $110
8 $130
10 $150
1. How much money will Jeanette have saved after 12 months?
2. How much money will Jeanette have saved after 24 months?
3. In how many months will Jeanette have saved $140? $190?
4. What is the rate of change?
5. How does knowing the rate of change help you in answering questions in this problem?
6. What equation would you write to represent this data?
Answers
I'm not that good at explaining but here are my answers:
1- 170
2-340
3- 9 months for $140 and 14 months for $190
4- 10
5-If the x (or months savings) goes up or down, so does the y (or account balance)
6- 70y +20n= 2x
explanation:
1- 150 + 20 = 170
2- 170*2= 340
3-the number always goes up by 20, if it's something like an uneven number, let's say 3, which is after 2, add 10.
4- can't explain this, I used a calculator
5- i don't think I have to explain
6- I don't know, I just took a random equation and "plugged in" the numbers from the equation I was working on.
Kelsey went to the donut shop before work. The ratio of chocolate donuts to sprinkled donuts that she bought is 3 to 4. If Kelsey bought a total of 21 donuts, how many of the donuts were sprinkled
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Answer:
she'd have 16 sprinkled donuts
Multiplying radicals. Simplify
Answers
Answer:
[tex]\sqrt[9]{10}[/tex]
Step-by-step explanation:
When you're multiplying radicals you start by multiplying inside the radical. Break it down so it's easier for you:
1. [tex]\sqrt{15}*\sqrt{6} = \sqrt{90}[/tex] beacause 15 * 6 = 90
2. The next step is to simplify that radical. You simplify by finding factors of that number. The factors of 90 are:
1 * 902* 453 * 305 * 186 * 159 * 10You want to take the biggest factors of that number (yours is 9 * 10). After that you are going to break it down more. What are the factors of 9 and what are the factors of 10? Well the biggest factors of 9 are 3 * 3 and the biggest factors of 10 are 5 * 2, so 10 cannot be broken down anymore because it doesn't have two of the same numbers so 10 is going to stay the same, but 9 has two of the same number (3). What we're left with is [tex]\sqrt[3]{10}[/tex] because the same numbers stay out of the radical and the single number stays inside.
3. We can't forget about the 3 that we left out, so what we're going to do is write [tex]3 *\sqrt[3]{10}[/tex]. When you multiply that out you keep the outside with the outisde which results in the answer being [tex]\sqrt[9]{10}[/tex] because this cannot be simplified anymore.
Identify the slope represented in the graph below.
Answers
The slope is -2
Explanation:
I choose two points on the graph (0,-2) and (-1,0) and put it into the slope form using two coordinate pairs which is y-y/x-x and got -2/1 which simplifies down to -2
What is the solution to the system of equations?
y = -2x – 6
4x - y = -36
A (15, 24)
B (-7, 8)
C (-5, 4)
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how I can answer this question, NO LINKS, if you answer correctly I will give u brainliest!
Answers
Answer:
It's a direct variation condition,
hence ratio remains constant.
39 - 3
x - 4
x = 4×39/3 = 4×13 = 52 .. (for first blank space)
39 - 3
x - 10
x = 39×10/3 = 13×10 = 130 .. (for second blank space)
Answer:
39 ____ 3
52 ____ 4
130 ____ 10
To get the smaller numbers to a bigger one you'll times them by 13.
Divide 39 by 3 and you'll get 13 then time 4 by 13 to get the missing box answer do the same for number 10.
Please help me with this problem I will give brainpower to the first person that answers this correctly with an explaination
Answers
Hi!
To solve this, we can set up an equation. If L, the length, is 3 less than 2 times W, the width, it would look like this. L = 2w - 3
Now, the perimeter equation is L + L + w + w = 48, so we plug in the L's and get this:
2w - 3 + 2w - 3 + w + w = 48
Now solve, add like terms:
6w - 6 = 48
add 6 to both sides
6w = 54
divide both sides by 6
w = 9
The width is 9, and the length is two times this minus 3, so 2(9) - 3 = 18 - 3 = 15
Therefore your width is 9 and your length is 15
Hope this helps! :D
HELP QUICK PLS Which property is used in the following expression? (4 + 8) + 6 = 4 + (8 + 6) Associatve Property Commutitive Property Distributive property
Answers
the answer is distrubutive property
Please help me with this!!! I will mark brainlist!!!
Answers
Answer: The value of Angle F = 47
Step-by-step explanation:
Write an equation of a line which has a slope of -2 and goes through the point (-4,11)
Answers
Answer:
See answers below
Step-by-step explanation:
slope-intercept form- y=-2x+3
point slope form- y-11=-2(x+4)
Use the given slope and point to substitute into the point-slope formula
then you can solve for y=mx+b
what’s an perpendicular equation of a line to y=-2/7x+9 that passes through the point (4,-6)
Answers
Answer:
has no prison pits as melatarea
answer this for 10 points
Answers
Answer:
1 2 5 and 6
Step-by-step explanation:
Cube root -125 over 216
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Hope this helps
3. Rich has t dollars in his checking account. On June 3, he deposited
w, h, and v dollars, and cashed a check for k dollars. Write an alge-
braic expression that represents the amount of money in his account
after the transactions.
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b = his end total after the transactions
Solve this math problem. Correct answer gets brainliest.
Answers
Answer:
$28.50
Step-by-step explanation:
21 ÷ 14 = 1.5
19 · 1.5 = 28.50
Hope this helps!!
PLEASE HELP! ILL GIVE BRAINLIEST!I! I uploaded a picture below.
Answers
Answer: The answer is 3 basic toppings and 2 premium toppings.
PLEASE HELP QUESTION IN PICTURE
Answers
Answer:
-3/2
Step-by-step explanation:
1) find the slope of (-9,-4), and (3,4)
4-(-4)=8, and 3-(-9)=12
the slope is 8/12 or 2/3 when simplified.
the PERPENDICULAR slope is the negative reciprocal of the slope, so
-3/2 is the answer.
what is 7x5 and ummmmmm
Answers
Answer:
35
Step-by-step explanation:
Answer:
35
Step-by-step explanation:
PLEASE HELP
5) Which function represents the relationship between x and y?
A) y = x + 2
B) y = 2x
C) y = x + 4
D) y = x-2.
Answers
Answer:
A) y=x+2
Step-by-step explanation:
We can see that each time x increase by 2, y also increases by 2. The slope of the line formed by this equation is therefore 1. But we start out with y = 2, even when x=0. That's the definition of the y-intercept: the value of y when x is equal to zero.
The slope-intercept form of a straight line equation is y = mx + b, where m is the slope (1, in this case) and b is the y-intercept (0 in this case).
We have y = 1x + 2, or y = x + 2
Name all points collinear with point E.
Answers
Answer:
n,c,b
Step-by-step explanation:
what is .9 repeating as a fraction
Answers
There is no answer to this because:
Any digit repeating like that would usually be ?/9
Example: 0.1 repeating = 1/9
0.2 repeating = 2/9
0.8 repeating = 8/9
But 0.9 repeating? There should not be a fraction for that.
If you want to round up to 1, you can.
What is the answer HELP!!!!!!!!!!!!!!!!!!!!!!!!!
Answers
You went to a carnival. Each ride cost $2.50. Let n represent the
number of rides.
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suppose+the+population+of+a+small+town+is+16000+and+growing+exponentially+at+2.2+%.+how+many+years+until+the+population+reaches+25000?
Answers
Answer:
20.5 years
Step-by-step explanation:
The population is given by the exponential function ...
p = 16000·(1 +0.022)^t
When p = 25000, we can find the value of t by taking logarithms.
log(25000/16000) = t·log(1.022) . . . . . divide by 16000, take the log
t = log(25/16)/log(1.022) ≈ 20.5 . . . . . . divide by the coefficient of t
It will take about 20.5 years for the population to reach 25000.
Help me Pleaseeeeeee!!!!!!!!!!!!
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Answer:
I don't know
Step-by-step explanation:
Can someone help me with this please?
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Which of the following expressions is equivalent to.
Answers
Answer:
the answer is this hope it helps