Base on definition of the inverse, f(g(x))=x and vice versa. Given f(x)=1/2x+3 and g(x)=2x-6, write a composition (s) should be used to prove that f(x) and g(x) are inverses of each other.
Answers
To verifiy if these are inverses we need to calculate the expression "f(g(x))" which uses the expression for "g(x)" in place of x on the expression of "f(x)". We have:
[tex]\begin{gathered} f(g(x))=\frac{1}{2}(2x-6)+3 \\ f(g(x))=\frac{2x}{2}-\frac{6}{2}+3 \\ f(g(x))=x-3+3 \\ f(g(x))=x \end{gathered}[/tex]Since the result is f(g(x)) = x, they are inverses of each other.
Related Questions
For 5 cups of milk you need 4 cups of oatmeal, how many cups of oatmeal will be needed for 1 cup of milk?
Answers
The ratio milk-oatmeal has to stay the same; then, if x represents the cups of oatmeal we need for 1 cup of milk, we get:
[tex]\begin{gathered} \frac{5}{4}=\frac{1}{x} \\ \Rightarrow x=\frac{4\cdot1}{5}=\frac{4}{5}=0.8 \end{gathered}[/tex]The answer is 4/5 or 0.8 cups of oatmeal
Find the circumference of a circle with a diameter of meters. Use as an approximation for . Round your answer to the nearest whole meter. Enter only the number.
Answers
Solution:
The circumference of a circle is expressed as
[tex]\begin{gathered} circumference=\pi\times d \\ where\text{ d}\Rightarrow diameter\text{ of the circle} \end{gathered}[/tex]Given that the diameter of the circle is 13 meters, we have
[tex]d=13[/tex]By substitution, we have
[tex]\begin{gathered} circmference=\pi\times13 \\ where\text{ }\pi=3.14 \\ thus, \\ circumference=3.14\times13 \\ =40.82 \\ \therefore \\ circumference\approx41\text{ meters} \end{gathered}[/tex]Hence, the circumference of the circle, to the nearest meter, is
[tex]41[/tex]Translate this phrase into an algebraic expression.Eight less than the product of 14 and Mai's heightUse the variable m to represent Mai's height.
Answers
Given the expression;
"Eight less than the product of 14 and Mai's height"
We shall represent Mai's height by letter m.
The product of 14 and Mai's height shall be;
[tex]\begin{gathered} 14\times m \\ =14m \end{gathered}[/tex]Eight less than this would now be written as;
[tex]\begin{gathered} 8\text{ less than 14m} \\ 14m-8 \end{gathered}[/tex]ANSWER:
[tex]14m-8[/tex]50 < 2x + 10 and 2x + 10 < 110 solve for x in this inequality
Answers
An equilateral triangle is folded in half. What is x, the height of the equilateral triangle? (G.8b, 1 point) AA 60° 600 14 cm O A. 14V3 cm OB. 14 cm O C. 7 cm O D. 773 cm
Answers
hi
do you have a picture of this question?
thanks
To solve this problem, we will use the trigonometric function tangent
tan 60 = height / 7
height = 7*tan 60
height = 12.12 cm it's letter D
The first three terms of an arithmetic sequence are as follows.19, 28, 37Find the next two terms of this sequence.19, 28, 37,
Answers
Answer:
46, 55
Explanation:
Given the arithmetic sequence:
[tex]19,28,37[/tex]We want to find the next two terms.
An arithmetic sequence is a sequence in which the next term is obtained by the addition or subtraction of a constant called the common difference.
First, we determine the common difference:
[tex]Common\;Difference=28-19=37-28=9[/tex]Thus, the next two terms are obtained below:
[tex]\begin{gathered} 4th\;term:37+9=46 \\ 5th\;term:46+9=55 \end{gathered}[/tex]The next two terms of the sequence are 46 and 55.
b A. Linear Function Does the graph represent a B. Neither C. Non-Linear Function linear function or a non-linear function?
Answers
It is a cubic function.
Joan's expenses for a month were: rent, $680; transportation,$265; food, $487; clothing, $95; utilities, $240; and otheritems, $55. What was the total for her expenses?
Answers
For finding out the total of Joan's expenses for a month, we need to add up all her expenses:
[tex]\text{Rent + Transportation + Food + Clothing + Utilities + Other items}[/tex]Now you can calculate the Joan's expenses, this way:
[tex]680+265+487+95+240+55\text{ }[/tex]Cara went to the store and bought 6 flowers,f and 3 boxes of candy, c, for gifts. Write this situation as an expression
Answers
6 flowers (f)
3 boxes of candy (c)
Add the products
6f+3c
translate the following expression into symbols: the whole numbers are a subset of the integers
Answers
Whole numbers: W
Integers: Z
the whole numbers are a subset of the integers:
To represent that an element is a subset of another elemet you use the symbol: ⊆
Then, you get the next:
[tex](W\subseteq Z)[/tex]Two similar pyramids have slant heights of 4 and 6.1) Find the scale factor.2) If the volume of the smaller pyramid is 48 meters cubed, what is the volume of the larger pyramid?
Answers
ANSWER
[tex]\begin{gathered} 1)\frac{3}{2}\text{ or 1.5} \\ 2)162m^3 \end{gathered}[/tex]EXPLANATION
1) The slant heights of the pyramids are 4 and 6.
The scale factor is the ratio of the corresponding sides of two figures, hence, the scale factor of the two pyramids is:
[tex]\begin{gathered} \frac{6}{4} \\ \Rightarrow\frac{3}{2}\text{ or 1.5} \end{gathered}[/tex]2) The ratio of the volumes of two similar figures is equal to the cube of their scale factor.
Let the volume of the bigger pyramid be p.
This means that:
[tex]\begin{gathered} \frac{p}{48}=(\frac{3}{2})^3 \\ \Rightarrow\frac{p}{48}=\frac{27}{8} \end{gathered}[/tex]Solve for p by cross-multiplying:
[tex]\begin{gathered} p=\frac{27\cdot48}{8} \\ p=162m^3 \end{gathered}[/tex]That is the volume of the larger pyramid.
Chloe is planning an event and purchasing floral arrangements. At Petals Plus, the arrangements cost $35.85 each, and the store charges $67.00 for delivery of the flowers to the event site. At Blooms, the arrangements cost $25.09 each, and the store charges $118.00 for delivery of the flowers to the event site.How many arrangements should she purchase from Blooms in order to get a better value? A. 3 B. 1 C. 5 D. 4
Answers
Each arrangement costs $35.85, plus $67.00 for delivery.
This can be expressed as
[tex]35.85a+67[/tex]Where a represents the arrangements.
Blooms.Each arrangement costs $25.09, plus $118.00 for delivery.
[tex]25.09a+118[/tex]Where a represents the arrangements.
Now, we have to form an expression to relate both expressions.
[tex]35.85a+67<25.09a+118[/tex]This expression will give us the number of arrangements needed to get a better offer from Blooms.-
[tex]\begin{gathered} 35.85a-25.09a<118-67 \\ 10.76a<51 \\ a<\frac{51}{10.76} \\ a<4.74 \end{gathered}[/tex]Notice that we have to get less than 4.74 arrangements to get a better deal with Blooms.
Therefore, the right answer is D. 4.estimate by rounding 32 * 478
Answers
Rounding is an aproximation. Find the most simple multiplication possible
Multiplication x2 is the most simple , after 1 multiplication
Use result. 2^ 5 = 32
So this means multiply 478 by 2 , five times
Duplicate in every step
So then
478x2 = 956
956x2= 1912
1912x2= 3824
3824x2= 7648
7648x2= 15296
37.5:15.625 simplfy to get 2.4:1
Answers
Given:
[tex]37.5:15.625[/tex]Required:
To simplify the given.
Explanation:
Consider
[tex]37.5:15.625[/tex][tex]\begin{gathered} =\frac{15.625}{37.5} \\ \\ =\frac{1}{2.4} \\ \\ =2.4:1 \end{gathered}[/tex]Final Answer:
[tex]37.5:15.625=2.4:1[/tex]Jonathan goes out for dinner with his family. They want to give the waiter a 20% tip. Which expression represents the total cost of the bill, including tip, if the cost of the dinner is d dollars? *
Answers
Tip percentage = 20% = 20/100 = 0.2 (decimal form)
Cost of the dinner = d
The total cost of the bill:
Multiply the cost of the dinner by (1+0.20).
D (1+0.20)
d 1.20
1.2 d
Please write [tex]4 {x}^{2} + 9 {y}^{2} - 24x + 18y + 9 = 0[/tex]in standard form
Answers
What ones are the right ones I’ve been trying for 40 minutes to get it and I can’t
Answers
Answer:
Explanations:
The rate of change of a line is also known as a slope. The fomula for calculating the slope of a line is gi
Answer not the equation just checking my answer to make sure it is right?
Answers
The given equation is:
[tex]A(n)=1400+(n-1)(0.02\cdot1400)[/tex]Substitute n=7 to get:
[tex]\begin{gathered} A=1400+6\times0.02\times1400 \\ A=1568 \end{gathered}[/tex]Option D is correct.
there are 100 cards in a hat numbered 1 to 100 the game is to draw one card out of the hat if the number you draw is odd you win $19.if the number you draw is even you win nothing. if you play the game what is the expected payoff
Answers
Answer:
Explanation:
There are 100 cards numbered 1 to 100.
• The number of even cards, E = 50
,• The number of odd cards, O = 50
The various probabilities are calculated below:
[tex]\begin{gathered} P(E)=\frac{50}{100}=\frac{1}{2} \\ P(O)=\frac{50}{100}=\frac{1}{2} \end{gathered}[/tex]Can you assist with #7 look ate the finished rest as a guide if necessary
Answers
We know that the segment DP is an angle bisector of angle BDC; this means that angle 1 has to be half angle BDC, that is:
[tex]m\angle1=\frac{1}{2}m\angle BDC[/tex]Plugging the expresions given for each angle we have:
[tex]\begin{gathered} 6x+6=\frac{1}{2}(15x) \\ 12x+12=15x \\ 15x-12x=12 \\ 3x=12 \\ x=\frac{12}{3} \\ x=4 \end{gathered}[/tex]Therefore, the value of x is 4
I need help knowing how the graph would look like with a polynomial with roots 4,-2 and 7 thanks!
Answers
Answer:
Explanation:
Here, we want to get the graph of the polynomial
To get that, we have the expression of the polynomial as follows:
[tex]\begin{gathered} if\text{ x = 4 , a linear factor is x-4} \\ x\text{ = -2, a linear factor is x+2} \\ x\text{ = 7 , a linear factor is x-7} \end{gathered}[/tex]Thus, we have the polynomial expression as follows:
[tex](x-4)(x+2)(x-7_)[/tex]Now, we can have the plot as shown below:
We have the y-intercept at y = 56
The table below represents the displacement of a horse from its barn as a function of timeTime(hours)Displacementfrom barn(feet)У08158210831584208Part A: What is the y-intercept of the function, and what does this tell you about the horse? (4 points)Part B: Calculate the average rate of change of the function represented by the table between x = 1 to x = 3 hours, and tell what the average rate represents. (4 points)Part C: What would be the domain of the function if the horse continued to walk at this rate until it traveled 508 feet from the barn? (2 points)(10 points)
Answers
Part A.
We are asked to find the y-intercept of the function. The y-intercept of a function is the value of y when the value of x is 0.
In this case, the y-intercept is given in the table:
When x is equal to 0, y is equal to 8. So the y-intercept is 8.
What this tells us about the horse is that he started his displacement when he was already 8 feet away from the barn.
Part B.
In this part, we need to find the average rate of change between x=1 and x=3. First, we label these limits as follows:
[tex]\begin{gathered} a=1 \\ b=3 \end{gathered}[/tex]And now, we will use the average rate of change formula:
[tex]\frac{f(b)-f(a)}{b-a}[/tex]In this case since a=1 and b=3:
[tex]\frac{f(3)-f(1)}{3-1}[/tex]f(3) is the value of the function (the value of y) when x is equal to 3. And from the table, we can see that it is 158:
[tex]f(3)=158[/tex]And we do the same for f(1), it is the value of y when x is equal to 1. And from the table, we see that it is 58:
[tex]f(1)=58[/tex]Going back and substituting these values into the average rate of change formulá:
[tex]\frac{158-58}{3-1}[/tex]Solving these operations:
[tex]\frac{100}{2}=50[/tex]The average rate of change is 50 feet/hour. This average rate of change represents the number of feet that the horse moves in 1 hour. It is the displacement of the horse per hour.
Part C.
We are asked to find the domain of the function if the horse continued to walk at this rate until it traveled 508 feet.
To solve this problem, we can use an equation for the relationship between hours and distance:
[tex]y=50x+8[/tex]From the table, we have that x are the hours and y is the distance. That is because the distance is the multiplication of the rate of change by the hours, and since the horse started at 8 feet, we need to add that to the function.
With this function we find the value of x (the hours) when the distance traveled is 508 feet:
[tex]508=50x+8[/tex]Solving for x:
[tex]\begin{gathered} 508-8=50x \\ 500=50x \\ \frac{500}{50}=x \\ 10=x \end{gathered}[/tex]At 508 feet the horse has traveled for 10 hours.
Finally, to solve this part C of the problem we need the domain.
The domain is defined as the values possible for x. In this case, those values go from 0 hours to 10 hours (as we just found 10 hours is the limit)
So, the domain is:
[tex]\lbrack0,10\rbrack[/tex]Or you can also write as:
[tex]0\leq x\leq10[/tex]Phyllis invested 54000 dollars, a portion earning a simple interest rate of 4% per year and the rest earning a rate of 7% per year. After one year the total interest earned on these investments was 3270 dollar. how much money did she invest at each rate
Answers
$54,000
a portion at simple interest 4%
another portion simple interest 7%
earned in one year $3,270
How much on each rate ?
Simple interest formula: P * r * t where P is the principal, r is the rate, and t is the time
Earnings at 4% = P1 * 0.04 * 1 = 0.04 P1
Earnings at 7% = P2 * 0.07 * 1 = 0.07 P2
Both add 3270, so we can write:
Equation 1: 0.04 P1 + 0.07 P2 = 3270
also both quantities invested add 54000, so we can write:
Equation 2: P1 + P2 = 54000
Solving Equation 2 for P1:
P1 = 54000 - P2
Using this value into equation 1:
0.04(54000 - P2) + 0.07 P2 = 3270
Solving for P2:
2160 - 0.04 P2 + 0.07 P2 = 3270
0.03 P2 = 3270 -2160 = 1110
P2 = 1110/0.03 = 37000
P2 = 37000
Using this value into the expression we found for P1:
P1 = 54000 - P2 = 54000 - 37000 = 17000
P1 = 17000
Answer:
She invested $17,000 at 4% and $37,000 at 7%
what must be added to 40 to make 10²
Answers
Answer:
Step-by-step explanation:
Since 10^2= 100, you just have to subtract 40 from that:
100-40= 60 as answer
100-40=60 this is the answer
I need help with this please?
Answers
SOLUTION
Write out the given information
[tex]\begin{gathered} \text{Initial volume of gas=40litre} \\ \text{percentage of gas lost per day=0.5\%} \\ \text{The percentage remaining will be } \\ 100-0.5=99.5\text{ \%} \end{gathered}[/tex]Then
After each day, the gas will be remaining
[tex]\begin{gathered} \frac{99.5}{100}=0.995 \\ \\ \end{gathered}[/tex]Then, then for one week, we will have
[tex](0.995)^7[/tex]Hence
The volume of the gas that will be remaining will be
[tex]40(0.995)^7[/tex]Therefore
The volume of gas after one week is 40(0.995)^7
The fourth option is correct (D)
Solve the following system of equations. (Hint: Use the quadratic formula.) f(x) = 2x² 3x g(x)=-3x² + 20 (0.-10) and (1, 17) (-2.2, 5.9) and (3.2, 0.9) (28.-3.0) and (-1.5, -1) (-2.2, 5.9) and (2.8, -3,0)
Answers
The solution of the system of equation is the intersection point of the two quadratic equations, so we need to equate both equations, that is,
[tex]2x^2-3x-10=-3x^2+20[/tex]So, by moving the term -3x^3+20 to the left hand side, we have
[tex]5x^2-3x-30=0[/tex]Then, in order to solve this equation, we can apply the quadratic formula
[tex]x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]In our case, a=5, b=-3 and c=-30. So we get
[tex]x=\frac{3\pm\sqrt{(-3)^2-4(5)(-30)}}{2(5)}[/tex]which gives
[tex]\begin{gathered} x=2.76779 \\ and \\ x=-2.16779 \end{gathered}[/tex]By substituting these points into one of the functions, we have
[tex]f(2.76779)=-2.982[/tex]and
[tex]f(-2.16779)=5.902[/tex]Then, by rounding these numbers to the nearest tenth, we have the following points:
[tex]\begin{gathered} (2.8,-3.0) \\ and \\ (-2.2,5.9) \end{gathered}[/tex]Therefore, the answer is the last option
It doesn’t have a box on the picture for an answer because that thing has an error, but I still need help with finding the solutions.
Answers
Answer:
[tex]\log _a(\frac{\sqrt[]{3}}{10})=-0.0410[/tex]Step-by-step explanation:
Given the logarithms, find the following expressions:
[tex]\log _a(\frac{\sqrt[]{3}}{10})=\log _a\sqrt[]{3}-\log _a10[/tex]Using the rule for root logarithms:
[tex]\begin{gathered} \log _a\sqrt[p]{x}=\frac{\log _ax}{p} \\ \text{Solving the expression:} \\ \frac{\log_a3}{2}-(\log _a5\cdot\log _a2)=\frac{0.6131}{2}-(0.8982\cdot0.3869) \\ =-0.0410 \end{gathered}[/tex]Find the equation of the parabola with the following properties. Express your answer in standard fornFocus at (1, -4)Directrix is the x-axisAnswer 2 Points
Answers
Answer:
[tex]y=-\frac{1}{8}(x-1)^2-2[/tex]First, to know the opening of the parabola, let us solve for p:
[tex]\begin{gathered} \text{ Focus:}(1,-4) \\ \text{ Directrix: y}=0 \\ p=\frac{0-(-4)}{2}=\frac{4}{2}=2 \end{gathered}[/tex]Now, the formula for the parabola is noted as:
[tex]y=\frac{1}{4p}(x-h)^2+k[/tex]Since our p is 2, the vertex of the parabola would be at:
[tex]\begin{gathered} v(1,-4+2) \\ v(1,-2) \end{gathered}[/tex]This will now be our (h,k).
With these, we know that
h = 1
k = -2
p = 2
We substitute these values to the equation:
[tex]\begin{gathered} y=\frac{1}{4p}(x-h)^{2}+k \\ y=\frac{1}{4(2)}(x-1)^2+(-2) \\ y=\frac{1}{8}(x-1)^2-2 \end{gathered}[/tex]Since our parabola is opening downward, we will add a negative sign in front of the equation.
The equation is, therefore:
[tex]y=-\frac{1}{8}(x-1)^2-2[/tex]Which of the following ratios is equivalent to 1/2?O1:2O1:1O2:1
Answers
A ratio of the form:
[tex]a\colon b[/tex]Can be written as:
[tex]\frac{a}{b}[/tex]Therefore:
[tex]\frac{1}{2}=1\colon2[/tex]I need help statements and reasonings for my 3rd and 4th line so it can be a complete validation
Answers
Answer:
Step-by-step explanation:
[tex]undefined[/tex]