Final answer:
Two functions are inverses if, when composed, they return the original input value (identity function). For example, an exponential function and its inverse, the natural log, undo each other. This can be checked with the equations by rearranging and setting equal to isolate constants or applying the functions one after the other.
Explanation:
How to Determine if Two Functions Are Inverses
To determine if two functions, F and G, are inverses of each other, you can compose the functions and see if the result is the identity function. The composition of two inverse functions will output the original input value, meaning that if you take a value x, apply function F to get F(x), and then apply function G to this result, i.e., G(F(x)), you should get x back if F and G are true inverses. The same must hold true in the reverse order, where G is applied first, followed by F: F(G(x)) = x.
To illustrate this concept, let's consider the exponential function and its inverse function, the natural log.Exponential functions are undone by their inverses, natural logs, according to the relation[tex]eln(x) = ln(ex)[/tex]to see this is with the equations involving a constant A, for instance, these can be rearranged to isolate ln A and set them equal to obtain an identity.
Another familiar context is the square and square root functions. To "undo" a square, you take the square root and vice versa. This "undoing" operation is a key characteristic of inverse functions.
Remember that not all functions have inverses, only those that are one-to-one, meaning for each x there is a unique y and vice versa, have inverses that can be defined over their entire domain.
How many solutions can be found for the linear equation? 10(x + 6) 2 - 9 = (15x + 1) 3 - 8
The given linear equation will have only one possible solution
What is function?A function is a relation between a dependent and independent variable. We can write the examples of functions as -
y = f(x) = ax + b
y = f(x, y, z) = ax + by + cz
Given is to find the total number of solutions for the linear equation -
10(x + 6) 2 - 9 = (15x + 1) 3 - 8
A linear equation is a equation of degree one. The number of solutions of a equation are equal to its degree. So, the given linear equation will have only one solution.
Therefore, the given linear equation will have only one possible solution.
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Steve’s car manual says his car does 42 mpg. A gallon is 4.55 litres. A litre of diesel costs £1.38. Steve drives from Leeds to Edinburgh a distance of 215 miles. How much would it cost for diesel
To calculate the cost of diesel for Steve's trip from Leeds to Edinburgh, divide the distance by the car's mpg to find the number of gallons needed. Then, multiply the gallons by the cost of diesel per gallon to get the total cost.
Explanation:To determine the cost of diesel for Steve's trip, we need to calculate the number of gallons he will need and then multiply by the cost per gallon. Steve's car does 42 miles per gallon (mpg), and he is driving a distance of 215 miles. Therefore, he will need 215/42 = 5.12 gallons of diesel. Since a gallon is 4.55 liters, Steve will need 5.12 * 4.55 = 23.36 liters of diesel. The cost of diesel per liter is £1.38, so the total cost for diesel will be 23.36 * £1.38 = £32.19.
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choose the expressions that are equal to 5.92+3.48
The longer leg of a right triangle is 1inch longer than the shorter leg. the hypotenuse is 9inches longer than the shorter leg. find the side lengths of the triangle.
The Pythagorean theorem is applied to a right triangle with the longer leg being one inch longer than the shorter leg and the hypotenuse being nine inches longer than the shorter leg. The side lengths of the triangle are found to be 5 inches, 6 inches, and 14 inches.
Explanation:The subject of this question is mathematics, specifically the part of geometry that deals with right triangles and the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (c), is equal to the sum of the squares of the other two sides (a and b), i.e., a² + b² = c².
In this problem, the longer leg (a) is represented as b = a + 1 and the hypotenuse (c) as c = a + 9. If you substitute these two equations into the Pythagorean theorem, you get (a + 1)² + a² = (a + 9)². Solving this equation gives a = 5 inches.
Substituting a = 5 inches into the expressions for b and c, we get b = 6 inches and c = 14 inches. Therefore, the sides of the triangle are 5 inches, 6 inches, and 14 inches.
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Megan is constructing the bisector of AB¯¯¯¯¯. She has already constructed two arcs as shown.
What should Megan do for her next step?
Use the straightedge to draw XY←→.
Place the point of the compass on point A and draw an arc, using AX as the width for the opening of the compass.
Place the point of the compass on point X and draw an arc, using AX as the width for the opening of the compass.
Use the straightedge to draw AX←→ and BX←→.
Kalahira is correct but the letters at the end might confuse you.
Correct answer:
Use the straightedge to draw XY←→.Please help me thank you! PLEASE SHOW ALL WORK TOO!
I have 4 questions.
What is the sum of 100.0 g and 0.01 g, expressed in scientific notation and written with the correct number of significant figures?
The sum of 100.0 g and 0.01 g, expressed in scientific notation and written with the correct number of significant figures is 1.0001 * 10².
What is sum?Sum is the output of the mathematical operation, Addition.
The sum of two numbers a and b is written as a +b.
The first number is 100.0 g
The second number is 0.01 g.
Scientific notation is the method used to write very small and large quantities.
It makes it easy to understand and interpret.
The rules followed while writing significant figures is that, the digits which are not zero is always significant, zeroes between two significant digits are significant.
The number of the form 0.0001 = 1 * 10⁻⁴,
and the number 45000000 is written as = 4.5 * 10⁷.
The sum of 100.0 g and 0.01 g has to be determined.
Let x represent the sum of the numbers, then
x = 100.00 + 0.01
x = 100.01
The sum is 1.0001 * 10²
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Is y=-13/5x-3 5y=3x-10 parallel, perpendicular, or neither?
What is the sum of the arithmetic series below 2+5+8+...+59?
Which statement is always true?
a. linear pairs of angles are congruent
b. vertical angles are supplementary
c. adjacent angles are complementary
d. adjacent linear pairs of angles are supplementary?
The statement that is always true is: D. adjacent linear pairs of angles are supplementary.
What are Linear Pairs of Angles?Linear pairs of angles are angles that lie on the straight line. The sum of a linear pair is always 180 degrees, thus, they are regarded as supplementary angles.
Therefore, the statement that is always true is: D. adjacent linear pairs of angles are supplementary.
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An Expression that represents 40% of a number is 40n
Using the graph attached below, what are the common difference, the general term equation, and the 12th term of the arithmetic sequence?
Hint: asubn = asub1 + d(n − 1), where asub1 is the first term and d is the common difference.
OPTIONS:
A) d = −3, asubn = 3 − 4n, asub12 = −45
B) d = 4, asubn = 5n − 4, asub12 = 56
C) d = −5, asubn = 4 − 5n, asub12 = −56
D) d = −4, asubn = 5 − 4n, asub12 = −43
Answer:
C
Step-by-step explanation:
Cause i'm a genius
David is playing a trivia game where he gains points for correct answers and loses points for incorrect answers. At the start of round 3 his score is −1500 points. During round 3 he answered five 1000 point questions correctly and three 500 points questions incorrectly. What is his score at the end of round 3?
David’s score at the end of round 3 is calculated by adding the net points gained during the round to his initial score, resulting in a final score of 2000 points.
Calculating David's Score
To determine David's score at the end of round 3, we need to account for both his correct answers and incorrect answers during the round.
David's initial score at the start of round 3 is -1500 points.
During round 3:
He answered five 1000-point questions correctly. Each correct answer adds 1000 points. So, 5 correct answers add:
5 × 1000 = 5000 points
He answered three 500-point questions incorrectly. Each incorrect answer subtracts 500 points. So, 3 incorrect answers subtract:
3 × 500 = 1500 points
Next, we calculate the net change in his score by adding the points gained and subtracting the points lost:
Net change = 5000 points (gained) - 1500 points (lost) = 3500 points
Finally, we add this net change to his initial score:
Final score = -1500 points (initial score) + 3500 points (net change) = 2000 points
Thus, David's score at the end of round 3 is 2000 points.
A classroom has 4 new boxes of chalk and 6 individual pieces of chalk in use. How many total pieces of chalk are in the classroom?
Write the first ten positive perfect-square integers
A perfect square is a figure that can be conveyed as the product of two identical integers.
The first ten positive perfect squares are the following:
1 = 1 x 1
4 = 2 x 2
9 = 3 x 3
16 = 4 x 4
25 = 5 x 5
36 = 6 x 6
49 = 7 x 7
64 = 8 x 8
81 = 9 x 9
100 = 10 x 10
The word ____ tells you that the relationship describes an equation?
For which pairs of function is (f x g ) (x) = 12x
Find the distance, in feet, a particle travels in its first 4 seconds of travel, if it moves according to the velocity equation v(t)= −t2 + 4 (in feet/sec).
55 over 3
16 over 3
16
12
Answer:
16 feet
Step-by-step explanation:
The relationships between displacement (position), velocity and acceleration are:
[tex]\boxed{\boxed{\begin{array}{c}\textbf{DISPLACEMENT (s)}\\\\\text{Differentiate} \downarrow\qquad\uparrow\text{Integrate}\\\\\textbf{VELOCITY (v)}\\\\\text{Differentiate}\downarrow\qquad\uparrow \text{Integrate}\\\\\textbf{ACCELERATION (a)}\end{array}}}[/tex]
To find the distance a particle travels in its first 4 seconds of travel, we first need to determine if the particle changes direction at any point during this time.
The instant(s) when the particle changes direction is when its velocity is zero. Therefore, set the velocity function v(t) to zero and solve for t:
[tex]\begin{aligned}v(t)&=0\\-t^2+4&=0\\-t^2&=-4\\t^2&=4\\t&=\pm 2 \end{aligned}[/tex]
So, the particle changes direction at t = 2 seconds in its first 4 seconds of travel. This means that to find the distance the particle travels in its first 4 seconds of travel, we need to integrate the velocity function over the two intervals [0, 2] and [2, 4] seconds to find the particle's displacement over these intervals.
The displacement of the particle in its first 2 seconds of travel is:
[tex]\begin{aligned}\displaystyle \int^2_0 (-t^2+4)\; \text{d}t&=\left[\dfrac{-t^{2+1}}{2+1}+4t\right]^2_0\\\\&=\left[-\dfrac{t^{3}}{3}+4t\right]^2_0\\\\&=\left(-\dfrac{(2)^{3}}{3}+4(2)\right)-\left(-\dfrac{(0)^{3}}{3}+4(0)\right)\\\\&=-\dfrac{8}{3}+8+0-0\\\\&=\dfrac{16}{3}\end{aligned}[/tex]
The displacement of the particle in its next 2 seconds of travel is:
[tex]\begin{aligned}\displaystyle \int^4_2 (-t^2+4)\; \text{d}t&=\left[\dfrac{-t^{2+1}}{2+1}+4t\right]^4_2\\\\&=\left[-\dfrac{t^{3}}{3}+4t\right]^4_2\\\\&=\left(-\dfrac{(4)^{3}}{3}+4(4)\right)-\left(-\dfrac{(2)^{3}}{3}+4(2)\right)\\\\&=-\dfrac{64}{3}+16+\dfrac{8}{3}-8\\\\&=-\dfrac{32}{3}\end{aligned}[/tex]
The negative value means that the particle is travelling in the opposite direction.
So, the particle travels 16/3 feet in its first 2 seconds of travel, changes direction at t = 2 seconds, and travels 32/3 feet in the opposite direction in the next 2 seconds of travel.
Therefore, the total distance the particle travelled is the sum of the absolute values of the two displacements:
[tex]\textsf{Distance}=\dfrac{16}{3}+\dfrac{32}{3}=\dfrac{48}{3}=16\;\sf feet[/tex]
So, the particle travels 16 feet in its first 4 seconds of travel.
The distance a particle travels in the first 4 seconds, moving according to the velocity equation v(t)= -t² + 4, is found by integrating the velocity function over that interval, which yields 5.33 feet.
Explanation:To find the distance a particle travels given its velocity equation v(t)= -t² + 4 (in feet/sec), we need to integrate the velocity function over the desired time interval.
Since velocity is the rate of change of position with respect to time, the integral of the velocity function from 0 to 4 seconds will give us the displacement of the particle in that time interval.
The integral of v(t) from 0 to 4 seconds can be computed as follows:
∫ v(t) dt from t=0 to t=4= ∫ (-t² + 4) dt from t=0 to t=4= [-t³/3 + 4t] from t=0 to t=4= [(-4³/3 + 4 × 4) - (0³/3 + 4 × 0)]= [(-64/3 + 16) - 0]= (-64 + 48)/3= -16/3= -5.33 feetHowever, the absolute value of -5.33 feet is needed, because distance must be positive. Hence, the particle travels 5.33 feet in the first 4 seconds.
Which equation has the solutions x=1+/-\sqrt 5? x2 + 2x + 4 = 0 x2 – 2x + 4 = 0 x2 + 2x – 4 = 0 x2 – 2x – 4 = 0
x^2 - 2x -4 =0
Using the quadratic formula -b +/- √b^2 - 4(ac) / 2a
Replace the letters with the values from the equation:
2 +/- √-2^2 -4*(1*-4) / 2*1
X = 2 +/- 2√5 / 2
x = 1 +/-√5
The answer is: x^2 - 2x -4 =0
Jaime wants to display her math test scores by using either a line plot or a stem and leaf plot. Her test scores are:
93, 95, 87, 90, 84, 81, 97, 98.
Which best explains what type of graph will better display the data?
a stem and leaf plot because the data can be grouped into sets of 10
a stem and leaf plot because each data point contains two digits
a line plot because there are only a few data points
a line plot because the numbers are all clustered near each other
Answer:
A
Step-by-step explanation:
hope it helps!!!!!!!
Answer:
A on edge 2020
Step-by-step explanation:
Seventeen less than four times a number is twenty-seven find the number
Twice a number is equal to negative four. Which equation could be used to find the number? 2n = 4 2n = -4n 2n - 4 2n = -4
John took 45 minutes to bicycle to his grandmother's house, a total of four kilometers. what was his speed in km/hr?
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Part A: Explain why the x-coordinates of the points where the graphs of the equations y = 2^x and y = 4^x−2 intersect are the solutions of the equation 2^x = 4^x−2. (4 points)
Part B: Make tables to find the solution to 2^x = 4^x−2. Take the integer values of x between −4 and 4. (4 points)
Part C: How can you solve the equation 2^x = 4^x−2 graphically? (2 points)
B. 2-x = 4x+3
x 2-x 4x+3
-3 5 -9
-2 4 -5
-1 3 -1
0 2 3
1 1 7
2 0 11
3 -1 15
The table shows that none of the integers from [-3,3] work because in no case does
2-x = 4x+3
To find the solution we need to rearrange the equation to the form x=n
2-x = 4x+3
2 -x + x = 4x + x +3
2 = 5x + 3
2-3 = 5x +3-3
5x = -1
x = -1/5
The only point that satisfies both equations is where x = -1/5
Find y: y = 2-x = 2 - (-1/5) = 2 + 1/5 = 10/5 + 1/5 = 11/5
Verify we get the same in the other equation
y = 4x + 3 = 4(-1/5) + 3 = -4/5 + 15/5 = 11/5
Thus the only actual solution, being the point where the lines cross, is the point (-1/5, 11/5)
C. To solve graphically 2-x=4x+3
we would graph both lines... y = 2-x and y = 4x+3
The point on the graph where the lines cross is the solution to the system of equations ...
[It should be, as shown above, the point (-1/5, 11/5)]
To graph y = 2-x make a table....
We have already done this in part B
x 2-x x 4x+3
_ __
-1 3 -1 -1
0 2 0 3
1 1 1 7
Just graph the points on a Cartesian coordinate system and draw the two lines. The solution is, as stated, the point where the two lines cross on the graph.
I hope I helped!
I will still trying to see if I can solve them another way that might be clearer.
PLEASE HELP FAST AND SHOW ALL WORK!
Is the line through points P(-8-10) and Q(-5,-12) perpendicular to the line through points R(9,-6) and S(17,-5)? Explain.
The lines through points P and Q and points R and S are not perpendicular.
To determine if the line through points P(-8, -10) and Q(-5, -12) is perpendicular to the line through points R(9, -6) and S(17, -5), we need to calculate the slopes of both lines and check if they are negative reciprocals of each other. The slope of a line (m) through two points (x1, y1) and (x2, y2) is given by the formula m = (y2 - y1) / (x2 - x1).
For line PQ:
mPQ = (-12 + 10) / (-5 + 8) = -2 / 3
For line RS:
mRS = (-5 + 6) / (17 - 9) = 1 / 8
Now, the product of the slopes of two perpendicular lines is -1. Let's check if the product of mPQ and mRS is -1:
mPQ * mRS = (-2 / 3) * (1 / 8) = -2 / 24 = -1 / 12
Since -1 / 12 is not equal to -1, the lines are not perpendicular. Therefore, the line through points P and Q is not perpendicular to the line through points R and S.
When adding decimals, use a zero as a placeholder so that both decimals have the same number of digits after their decimal points?
Using a zero as a placeholder so that both decimals ahve the same number of digits after their decimal point is a good strategy while adding decimals because it helps you line the decimals up in the correct way to get the correct sum.
Reza paid $4,500 as a down payment on a car. She then made equal monthly paments of $250. Reza paid a total of $10,500 for her car. How many months did she make payments on the car and which fuction can be used to find the answer?
Reza made monthly payments for 24 months to pay off her car after making a $4,500 down payment. The total cost of the car was $10,500, and she paid the remaining amount in installments of $250 per month.
Explanation:Reza's car payments can be calculated using a simple algebraic function. She made an initial down payment of $4,500 and paid off the remaining balance with equal monthly payments of $250. The total cost of the car was $10,500. To find out how many months Reza made payments on the car, you can use the following function:
Total Cost = Down Payment + (Monthly Payment × Number of Months)
First, we subtract the down payment from the total cost:
$10,500 - $4,500 = $6,000
This is the amount Reza paid in monthly installments. Now we divide this amount by the monthly payment amount to find the number of months:
$6,000 / $250 = 24 months.
Therefore, Reza made payments for 24 months.
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Rose bought 7/20 kilogram of ginger candy and 0.4kilogram of cinnamon candy. Which did she buy more if
Final answer:
After converting 7/20 to a decimal, we find that Rose bought 0.35 kilogram of ginger candy, which is less than the 0.4 kilogram of cinnamon candy; therefore, Rose bought more cinnamon candy.
Explanation:
Rose bought 7/20 kilogram of ginger candy and 0.4 kilogram of cinnamon candy. To determine which she bought more of, we need to compare these amounts. The fraction 7/20 can be converted into a decimal to make the comparison easier. To convert a fraction to a decimal, you divide the numerator by the denominator.
So, 7 ÷ 20 = 0.35. This means that Rose bought 0.35 kilogram of ginger candy, which is less than the 0.4 kilogram of cinnamon candy. Therefore, Rose bought more cinnamon candy than ginger candy.
A test has a mean of 80 and standard deviation of 4. what score would be 1 deviation from the mean
Test has a mean of 80 and standard deviation of 4 than after one deviation from the mean, most of the score will fall in the interval ([tex]\mu - \sigma\;,\; \mu + \sigma[/tex]) which is (76 , 84).
Given :
Mean, [tex]\mu = 80[/tex]
Standard Deviation, [tex]\sigma = 4[/tex]
After one deviation from the mean, most of the score will fall in the interval ([tex]\mu - \sigma\;,\; \mu + \sigma[/tex]). Where [tex]\mu[/tex] is the mean which is arithmetic average of all values and [tex]\sigma[/tex] is the standard deviation which is the square root of its variance.
Now, the value of [tex]\mu - \sigma = 80-4 = 76[/tex].
And the value of [tex]\mu + \sigma = 80+4=84[/tex].
After one deviation from the mean, most of the score will fall in the interval ([tex]\mu - \sigma\;,\; \mu + \sigma[/tex]) which is (76 , 84).
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This graph shows the cost of buying dried fruit.
What is the slope of the line and what does it mean in this situation?
Select from the drop-down menus to correctly complete each statement.
The slope of the line is ____. This means that every of dried fruit costs $____.
the correct answer is =
The slope of the line is 3.5
This means that every pound
of dried fruit cost $3.50
Slope of the line is 3.5. This means that every of dried fruit costs $3.5 per lb.
From the graph attached,
Points shown on the graph are (4, 14) and (8, 28).
Since, slope of a line passing two points [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex] is given by,
Slope = [tex]\frac{y_2-y_1}{x_2-x_1}[/tex]
Therefore, slope of the line passing through (4, 14) and (8, 28) will be,
Slope = [tex]\frac{28-14}{8-4}[/tex]
= [tex]\frac{14}{4}[/tex]
= 3.5
This means that every of dried fruit costs $3.5 per lb.
Therefore, slope of the line is 3.5. This means that every of dried fruit costs $3.5 per lb.
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