Answer Before defining the inverse of a function we need to have the right mental image of function.
Consider the function f(x) = 2x + 1. We know how to evaluate f at 3, f(3) = 2*3 + 1 = 7. In this section it helps to think of f as transforming a 3 into a 7, and f transforms a 5 into an 11, etc.
Now that we think of f as "acting on" numbers and transforming them, we can define the inverse of f as the function that "undoes" what f did. In other words, the inverse of f needs to take 7 back to 3, and take -3 back to -2, etc.
Let g(x) = (x - 1)/2. Then g(7) = 3, g(-3) = -2, and g(11) = 5, so g seems to be undoing what f did, at least for these three values. To prove that g is the inverse of f we must show that this is true for any value of x in the domain of f. In other words, g must take f(x) back to x for all values of x in the domain of f. So, g(f(x)) = x must hold for all x in the domain of f. The way to check this condition is to see that the formula for g(f(x)) simplifies to x.
g(f(x)) = g(2x + 1) = (2x + 1 -1)/2 = 2x/2 = x.
This simplification shows that if we choose any number and let f act it, then applying g to the result recovers our original number. We also need to see that this process works in reverse, or that f also undoes what g does.
f(g(x)) = f((x - 1)/2) = 2(x - 1)/2 + 1 = x - 1 + 1 = x.
Letting f-1 denote the inverse of f, we have just shown that g = f-1.
Definition:
Let f and g be two functions. If
f(g(x)) = x and g(f(x)) = x,
then g is the inverse of f and f is the inverse of g.
Exercise 1:
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Finding Inverses
Example 1. First consider a simple example f(x) = 3x + 2.
The graph of f is a line with slope 3, so it passes the horizontal line test and does have an inverse.
There are two steps required to evaluate f at a number x. First we multiply x by 3, then we add 2.
Thinking of the inverse function as undoing what f did, we must undo these steps in reverse order.
The steps required to evaluate f-1 are to first undo the adding of 2 by subtracting 2. Then we undo multiplication by 3 by dividing by 3.
Therefore, f-1(x) = (x - 2)/3.
Steps for finding the inverse of a function f.
Replace f(x) by y in the equation describing the function.
Interchange x and y. In other words, replace every x by a y and vice versa.
Solve for y.
Replace y by f-1(x).
Example 2. f(x) = 6 - x/2
Step 1 y = 6 - x/2.
Step 2 x = 6 - y/2.
Step 3 x = 6 - y/2.
y/2 = 6 - x.
y = 12 - 2x.
Step 4 f-1(x) = 12 - 2x.
Step 2 often confuses students. We could omit step 2, and solve for x instead of y, but then we would end up with a formula in y instead of x. The formula would be the same, but the variable would be different. To avoid this we simply interchange the roles of x and y before we solve.
Example 3. f(x) = x3 + 2
This is the function we worked with in Exercise 1. From its graph (shown above) we see that it does have an inverse. (In fact, its inverse was given in Exercise 1.)
Step 1 y = x3 + 2.
Step 2 x = y3 + 2.
Step 3 x - 2 = y3.
(x - 2)^(1/3) = y.
Step 4 f-1(x) = (x - 2)^(1/3).
Exercise 3:
Graph f(x) = 1 - 2x3 to see that it does have an inverse. Find f-1(x). Answer
Step-by-step explanation:
pls brain list
Find 50% of 36:
Find 25% of 36:
Find 75% of 36:
Answer:
18, 9, 27
Step-by-step explanation:
36 ÷ 2 = 18. Dividing by 2 is the same as multiplying by 50%.
36 ÷ 4 = 9. Dividing by 4 is the same as multiplying by 25%.
Since 25% • 3 = 75%, you can do 9 (from previous) • 3 to get 27.
The amusement park won’t allow kids to ride roller coasters until they are a minimum of 48 inches tall. Michael is now 42 inches tall, and grows about 1/2 inches every month. How many months will it be until Michael can ride a roller coaster? (Solve with inequalities and solve for x)
42+1/2x=48
1/2x=6
x=6*2
x=12
12 months or a year until he could ride a roller coaster.
Help me with my math **50 pts*** I mark brainliest**
Answer:
Sure. Now what do you need help on? Please post the question.
Step-by-step explanation:
$$ \sqrt{6\frac{1}{4}} $$
2x+2 (1-4X) = 88-5 (x+4)7x + 2
Answer:
x=4/7
Step-by-step explanation:
what Is the answer to 1/6 + 2/3
Answer:
5/6
Step-by-step explanation:
When adding fractions, you must ensure the denominator is the same in both fractions.
In this case, the 3 can be multiplied by 2 to equal 6, the other denominator.
When multiplying fractions to create a common denominator, you must multiply the both the numerator and the denominator by the same value, to ensure that the fraction is still equivalent.
2/3 × 2/2 = (2×2)/(3×2) = 4/6
Replace 2/3 with its equivalent 4/6.
Now you will add the numerators together.
1/6 + 4/6 = (1+4)/6 = 5/6
Your final answer is 5/6
Answer:
5/6
Step-by-step explanation:
2/3--> 4/6 (GCF)
4/6+1/6=5/6
:)
A box contains 2 red marbles 3 green marbles and 3 blue marbles if we choose a marble then another marble without putting the first one back in the box what is the probability that the first marvel will be green and the second will be blue
Answer: 9/56
Step-by-step explanation:
Multiply (3a+2)(3a-2)
-26 =3v +1 can i please get a answer
Step-by-step explanation:
-26=3v+1
-1 on both sides
-27=3v
divide by 3 both sides
-9=v
1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20+21+22+23+24+25+26+27+28+29+30=
who get the first is question?
Answer: 465
Step-by-step explanation: thx for the free points :)
Answer:
Solution:
The greatest of Winnie's even numbers is $2\times 30 = 60,$ so her sum is
\[2+4+6+\cdots + 58 + 60.\]For Grogg's sum, there are five odd numbers between $0$ and $-10$, five between $-10$ and $-20$, and so on, so Grogg's sum is
\[(-1)+(-3)+(-5)+\cdots+(-57)+(-59).\]
We could compute these sums separately and find their difference, but instead, we'll look for a clever way to regroup to compute the difference
\[(2+4+6+\cdots +58+60)-((-1)+(-3)+(-5)+\cdots +(-57)+(-59)).\]Noting that Winnie's list steps up by 2 at each term while Grogg's steps down by 2 at each step, we write Winnie's list backwards, so that each list goes down by 2 at each step:
\[(60+58+56+\cdots + 4+2) - ((-1)+(-3)+(-5)+\cdots +(-57)+(-59)).\]Now, we see that the difference between a term in Winnie's backwards list and the corresponding term in Grogg's list is always the same:
\begin{align*}
60-(-1) &= 61,\\
58-(-3) &= 61,\\
56-(-5) &=61,\\
\vdots&\\
2-(-59) &= 61.
\end{align*}There are 30 such differences, so Winnie's sum is $30\times 61 = \boxed{1{,}830}$ greater than Grogg's sum.
Step-by-step explanation:
-4 > -2n
What could n be??
Answer:
Isolate the variable by dividing each side by factors that don't contain the variable.
Inequality Form:
n>2
Interval Notation:
(2,∞)
Write the equation
Show how to solve
Diego said that the answer to the question "How many groups of ¾ are in 1?" is 4/3 or 1 1/3.
Do you agree with him? Explain...
Answer:
Step-by-step explanation:
Yes. 1 ÷ ¾ = 1 × 4/3 = 4/3 = 1⅓
the question is what has BLM Done?
Answer:
Well, BLM has done a lot.
It brought people together, but also shredded others from each other.
The BLM was like a wilting rose still shining as bright as ever.
It started riots sure, but some BLM supporters were quite peaceful.
It all depends on how we look towards the BLM.
Which expressions are equivalent to 7(-3/4x-3)? Select two options.
Answer:
7(-3/4x-3) is the same as
7(-3/4x)-7(3) and
-5 1/4x-21
Step-by-step explanation:
Answer:
A and D
Step-by-step explanation:
got 100 on the quiz
Graph y > - 3/4x + 3
Answer:
Here's what the graph would look like:
You just graph it like a normal line, but make sure the line is dashed because you are looking for all greater values, not greater than or equal to.
Shade the area above the line because it is greater.
Plzz answer all odd questions if not all plzzz do at least 4 and show work. I will give brainliest to the one with the most correct questions answered with work shown this all my points so plzz answer correctly
Three-fifths of the members of the Spanish club are girls. There are a total of 30 girls in the Spanish club.
Which statements can be used to solve for x, the total number of members in the Spanish club? Select three options.
Three-fifths x = 30
Three-fifths = StartFraction x over 30 EndFraction
Three-fifths (Five-thirds) x = 30 (five-thirds)
(Three-fifths (five-thirds) x = 30 (five-thirds)
x = 50
Answer:
a d and e
Step-by-step explanation:
Answer:
A, D , E
Step-by-step explanation:
Just to concur with the person above this is the correct answer on edge
Pls do some research but can I bring a bearded dragon on a 12 hour car trip without any heath issues happening?
Answer:
after doing a little bit of research. NO your bearded dragon will not be affected as long as you feed it and keep it on a level surface so the enclosure it's in does not tip at all other than that it will be perfectly fine.
A 15 inch string is cut. One piece is 7.2 in. Which equation shows how to find the 2nd length?
how did you get the answer i need the fast hurry please
Answer:
7.2+x=15
x=15-7.2
x=7.8
Name the quadrant or axis where each point lies.
Answer:
(-1,0): x axis and 2nd quadrant, (3,-3): fourth quadrant, (-4,-3): 2nd quadrant, and (7,6): first quadrant
Step-by-step explanation:
Answer:
2nd,4th,2nd and 1st in quadrant they lie
Can someone help me with these two problems please?
Answer:
b
j
................................
You place two wooden support beams under a shelf, as shown. Find the value of x.
(2x + 10)
(x + 5)
Answer:
X=55
Step-by-step explanation:
(2x+10)+(x+5)=180
Combine like terms.
3x+15=180
Subtract 15.
3x=165
Divide by 3.
x=55
Answer:
see attached
Step-by-step explanation:
angles on a straight line
8 1/5-2 7/8 Fractions math
Answer:
5 13 /40
Step-by-step explanation:
hope this helps :))
EXTRA CREDIT: Complete the truth table. Choices are on the next page. (3 points)
Hurry please
Answer:
Step-by-step explanation:
Estimate the difference
5/9 - 3/8
Answer:
Este es el resultado de la preguntas 0.181555556
Answer:
13/72 i hope thats what your looking for
Step-by-step explanation:
What is 5/9 - 3/8?
What is 5/9 - 3/8?
Here's how to subtract 3/8 from 5/9:
5
9
−
3
8
Step 1
We can't subtract two fractions with different denominators. So you need to get a common denominator. To do this, you'll multiply the denominators times each other... but the numerators have to change, too. They get multiplied by the other term's denominator.
So we multiply 5 by 8, and get 40.
Then we multiply 3 by 9, and get 27.
Next we give both terms new denominators -- 9 × 8 = 72.
So now our fractions look like this:
40
72
−
27
72
Step 2
Since our denominators match, we can subtract the numerators.
40 − 27 = 13
So the answer is:
13
72
Step 3
Last of all, we need to simplify the fraction, if possible. Can it be reduced to a simpler fraction?
Video Player is loading.
To find out, we try dividing it by 2...
Nope! So now we try the next greatest prime number, 3...
Nope! So now we try the next greatest prime number, 5...
Nope! So now we try the next greatest prime number, 7...
Nope! So now we try the next greatest prime number, 11...
Nope! So now we try the next greatest prime number, 13...
Nope! So now we try the next greatest prime number, 17...
No good. 17 is larger than 13. So we're done reducing.
There you have it! The final answer is:
13/72
What is the approximate unit price of the Bacon Snacks?
A. $0.65 per pound
B. $1.53 per pound
O c. $2.29 per pound
O D. $7.99 per pound
Answer:
D. 7.99 perpound
Between 10 P.M. and 7:45 A.M., the water level in a swimming pool decreased by 3/8 inches . Assuming that the water level decreased at a constant rate, how much did the water level drop each hour?
Answer:
0.03846154... inches/hour
Step-by-step explanation:
between 10 P.M. and 7:45 A.M. = 9.75 hours
(3/8) inches / 9.75 hours = 0.03846154... inches per hour
EDIT: in fraction form
Answer:
1/26 inches/hour
Step-by-step explanation:
between 10 P.M. and 7:45 A.M. = 9 and 3/4 hours, or 39/4 hours
(3/8) inches / (39/4) hours = 1/26 inches per hour
12: 5 is equivalent to?
Geometry Help!
Please only answer if you know the answer. Look at the image below to answer the question.
Be sure to show all of your work and explain your thinking.
I posted 3 of these, so if you understand it, please go and check them out.
If you do not know the answer use the comment section, don't answer my question with a comment. Thanks, Good Luck!
Answer:
x = 88.76°Step-by-step explanation:
Use the law of cosines:
cos C = (a² + b² - c²)/(2ab)Apply to the given triangle and solve for x:
cos x = (52² + 48² - 70²)/(2*52*48)cos x = 108/4992cos x = 0.02163461538x = arccos 0.02163461538x = 88.76°