The true statement about a function and its inverse is, the range of both f(x) and f⁻¹(x) is all real numbers.
What is an inverse function?The inverse of a function f is denoted by f⁻¹ and it exists only when f is both one-one and onto function.
f⁻¹ isn't the reciprocal of f. The composition of the function f and the reciprocal function f⁻¹ gives the domain value of x.
The true statement about a function and its inverse is,
The range of both f(x) and f⁻¹(x) is all real numbers.
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Two friends are renting a property with a monthly rent of $975.00. The total square footage is 1,225 square feet, the first bedroom is 150 square feet, and the second bedroom is 130 square feet. Determine how much the friend with the smaller bedroom will pay for rent if they split the cost based on the square footage of each bedroom. (2 points)
$479.50
$481.35
$495.50
$501.35
Answer:
(a) $479.50
Step-by-step explanation:
You want to know how much the friend with the smaller bedroom will pay for rent if they split the $975 cost based on the square footage of each bedroom. The two bedrooms in the 1225 square foot property are 130 and 150 square feet.
SplitApparently, the cost of the non-bedroom area is to be split evenly. That common area is ...
1225 ft² -(150 +130) ft² = 945 ft²
Then the proportion due from the occupant of the smaller bedroom is ...
(945/2 +130)/1225 = 241/490 ≈ 0.49184
RentThat fraction of the rent is ...
0.49184 · $975 = $479.54
The nearest answer choice is $479.50.
__
Additional comment
If we were to assume the split is based only on bedroom size, then the rent paid for the 130 ft² space would be (13/28)($975) ≈ $452.68. This would be equivalent to splitting the rent for the common area in the same proportion as for the bedrooms.
We note that half the rent is $487.50, a number less than either of the last two answer choices. Those can be rejected for that reason. (The smaller bedroom is expected to pay less than half of the rent.)
What number makes the expressions equivalent? Enter your answer in the box.
12 (–1.4m + 0.4) =
m + 0.2
The number makes the expression equivalent is m = 0.2584.
What is a numerical expression?A numerical expression is a mathematical statement written in the form of numbers and unknown variables. We can form numerical expressions from statements.
Given, An expression 12(- 1.4m + 0.4) = m + 0.2.
- 16.8m + 4.8 = m + 0.2.
- 16.8m - m = 0.2 - 4.8
- 17.8m = - 4.6.
m = (-4.6/-17.8).
m = 0.2584.
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The vertex form of the equation of a horizontal parabola is given by x=1/4p(y-k)2+h , where (h, k) is the vertex of the parabola and the absolute value of p is the distance from the vertex to the focus, which is also the distance from the vertex to the directrix. You will use GeoGebra to create a horizontal parabola and write the vertex form of its equation. Open GeoGebra, and complete each step below.
Part A
Mark the focus of the parabola you are going to create at F(-5, 2). Draw a vertical line that is 8 units to the right of the focus. This line will be the directrix of your parabola. What is the equation of the line?
Part B
Construct the line that is perpendicular to the directrix and passes through the focus. This line will be the axis of symmetry of the parabola. What are the coordinates of the point of intersection, A, of the axis of symmetry and the directrix of the parabola?
Part C
Explain how you can locate the vertex, V, of the parabola with the given focus and directrix. Write the coordinates of the vertex.
Part D
Which way will the parabola open? Explain.
Part E
How can you find the value of p? Is the value of p for your parabola positive or negative? Explain.
Part F
What is the value of p for your parabola?
Part G
Based on your responses to parts C and E above, write the equation of the parabola in vertex form. Show your work.
Part H
Construct the parabola using the parabola tool in GeoGebra. Take a screenshot of your work, save it, and insert the image below.
Part I
Once you have constructed the parabola, use GeoGebra to display its equation. In the space below, rearrange the equation of the parabola shown in GeoGebra, and check whether it matches the equation in the vertex form that you wrote in part G. Show your work
Part J
To practice writing the equations of horizontal parabolas, write the equations of these two parabolas in vertex form:
focus at (4, 3), and directrix x = 2
focus at (2, -1), and directrix x = 8
Answer:
A. x = 3
B. A = (3, 2)
C. V = (-1, 2)
D. Left
E. See below.
F. p = -4
[tex]\textsf{G.} \quad x=-\dfrac{1}{16}(y-2)^2-1[/tex]
H. See attachment.
[tex]\textsf{J.} \quad x=\dfrac{1}{4}(y-3)^2+3[/tex]
[tex]x=-\dfrac{1}{12}(y+1)^2+5[/tex]
Step-by-step explanation:
Part AThe given focus of the parabola is F (-5, 2).
The equation for a vertical line is x = a.
Therefore, a vertical line that is 8 units to the right of the focus is:
[tex]\implies x=-5+8=3[/tex]
Therefore:
Directrix: x = 3Part BA line that is perpendicular to the directrix is a horizontal line.
The equation for a horizontal line is y = a.
As the horizontal line passes through the focus:
Axis of symmetry: y = 2The point of intersection, A, of the axis of symmetry and the directrix is:
(3, 2)Part CThe vertex of the parabola is the point that is halfway between the focus and the directrix.
Therefore:
Vertex = (-1, 2)Part DThe parabola will open sideways as the axis of symmetry is horizontal.
As a parabola never touches its directrix, this parabola will open to the left.
Part EThe absolute value of p is the distance between the vertex and the focus, and the distance between the vertex and the directrix.
For a sideways parabola, if p > 0 the parabola opens to the right, and if p < 0 the parabola opens to the left.
Part FSince the focus and directrix are 8 units apart, and the parabola opens to the left:
p = -4Part GVertex form of a sideways parabola:
[tex]\boxed{x=\dfrac{1}{4p}(y-k)^2+h}[/tex]
Given:
Vertex = (-1, 2) ⇒ h = -1, k = 2p = -4Substitute the values into the formula:
[tex]\implies x=-\dfrac{1}{16}(y-2)^2-1[/tex]
Part HSee attachment.
Part IRearrange the equation of the parabola shown in the attachment:
[tex]y^2+16x-4y=-20[/tex]
[tex]\implies 16x=-y^2+4y-20[/tex]
[tex]\implies 16x=-(y^2-4y+4)-16[/tex]
[tex]\implies 16x=-(y-2)^2-16[/tex]
[tex]\implies x=-\dfrac{1}{16}(y-2)^2-1[/tex]
This matches the equation in vertex form in part G.
Part JGiven:
Focus = (4, 3)Directrix: x = 2Therefore:
Vertex = ((4+2)/2, 3) = (3, 3)Parabola opens to the right, so p > 0.p = |4 - 3| = 1Therefore, the equation of the parabola with the given parameters is:
[tex]\boxed{x=\dfrac{1}{4}(y-3)^2+3}[/tex]
Given:
Focus = (2, -1)Directrix: x = 8Therefore:
Vertex = ((8+2)/2, -1) = (5, -1)Parabola opens to the left, so p < 0.p = -|2 - 5| = -3Therefore, the equation of the parabola with the given parameters is:
[tex]\boxed{x=-\dfrac{1}{12}(y+1)^2+5}[/tex]
pleaseeee help lol, will mark brainliest
Transforming the function f(x) = 2x by compression of a factor of 1/4 gives g(x) = 1/2x
Transformation of a FunctionTransformation of functions means that the curve representing the graph either "moves to left/right/up/down" or "it expands or compresses" or "it reflects". For example, the graph of the function f(x) = x2 + 3 is obtained by just moving the graph of g(x) = x2 by 3 units up. Function transformations are very helpful in graphing the functions just by moving/expanding/compressing/reflecting the curve without actually needing to graph it from scratch.
In the given function;
f(x)= 2x;
After a compression by a factor of 1/4, the function becomes
g(x) = 2x * 1/4 = 1/2x
The compressed function g(x) is 1/2x
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30 Points! Tell me everything you know about congruent polygons. How can we prove that triangles are congruent?
The polygons that are congruent are those whose vertices and sides line up perfectly. The 4 triangle congruence theorems that help in finding if the triangles are congruent or not are SSS, SAS, ASA, AAS, and RHS.
Define congruent polygons.The polygons that are congruent are those whose vertices and sides line up perfectly. Congruent polygons are polygons having the same size and shape, even if they can be rotations, translations, or mirror reflections of one another. This is another way to conceive of the idea.
A triangle's congruence can be established using the triangle congruence theorem or triangle congruence criterion. The definition of congruent is "absolutely equivalent in shape and size regardless of rotation, flipping, or turning." Congruent figures are described in geometry as shapes that are superimposed on one another, such as triangles and quadrilaterals.
The following are the triangle congruence criteria that aid in demonstrating triangle congruence.
SSS (Side, Side, Side)SAS (Side, Angle, Side)ASA (Angle, Side, Angle)AAS (Angle, Angle, Side)RHS (Right angle-Hypotenuse-Side or the Hypotenuse Leg theorem)To know more about congruency, visit:
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What is the image of the point (-6, 2) after a rotation of 180° counterclockwise
about the origin?
Answer:
[tex](6,2)[/tex]
Step-by-step explanation:
When rotating 180° about the origin, [tex](x,y) \longrightarrow (-x, -y)[/tex].
find the slope of the line passing through the points (-6,2) and (-7, 7).
Answer:
y-2=7(x+6)
Step-by-step explanation:
You got it there bro
Which expression is equivalent to (7x3)?(x8)2?
A
14x10
B
49x10
C
14x'
D 49x’
The expression equivalent to 7x³ × (x⁸) × 2 is 14x¹¹.
How to find an equivalent expression?Two expressions are said to be equivalent if they have the same value irrespective of the value of the variable(s) in them.
In other words, two algebraic expressions are said to be equivalent if their values obtained by substituting the values of the variables are the same.
Therefore,
7x³ × (x⁸) × 2
Using the law of indices,
[tex](x^{a}) (x^{b} ) = x^{a+b}[/tex]
Hence,
7x³ × (x⁸) × 2 = 7x³ × 2x⁸
7x³ × 2x⁸ = 14x¹¹
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PLEASE HELP ASAP!!!! FULL ANSWER ONLY!!!
A person standing close to the edge on top of a 72-foot building throws a ball vertically upward. The quadratic function h = − 16 t^2 + 84 t + 72 models the ball's height above the ground, h , in feet, t seconds after it was thrown.
a) What is the maximum height of the ball? ____________ feet
b) How many seconds does it take until the ball hits the ground? ____________ seconds
a) The ball reaches a maximum height of 182.25 feet.
b) The ball has a travel time of 6 seconds.
How to find the main features of a free fall case by analyzing quadratic equations
In this case we find a kind of uniform accelerated motion known as free fall, in which a particle is accelerated uniformly by gravity and any effect of other forces are neglected. The height of the ball in terms of time is described by a quadratic equation, whose definition is shown below:
h(t) = h' + v' · t + 0.5 · g · t²
Where:
h' - Initial height, in feet.v' - Initial velocity, in feet per second.t - Time, in seconds.g - Gravitational acceleration, in feet per square second.First, write the complete the equation of the height of the ball:
h(t) = - 16 · t² + 84 · t + 72
a) Second, clear the expression on one side and use the discriminant of the quadratic formula:
- 16 · t² + 84 · t + [72 - h(t)] = 0
84² - 4 · (- 16) · [72 - h(t)] = 0
7056 + 64 · [72 - h(t)] = 0
7056 + 4608 - 64 · h(t) = 0
11664 - 64 · h(t) = 0
h(t) = 729 / 4 ft
h(t) = 182.25 ft
The maximum height of the ball is 182.25 feet.
b) Third, use the quadratic formula to determine the roots of the quadratic equation:
t = - 84 / [2 · (- 16)] ± [1 / [2 · (- 16)]] · √[84² - 4 · (- 16) · 72]
t = 2.625 ± (1 / 32) · 108
t₁ = 6 s and t₂ = -0.75 s
The travel time of the ball is 6 seconds.
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Given f(x)=2x-2, find f(-3)
PLEASE HELP!!!!
Answer:4
Step-by-step explanation:
replace x with -3
f(x) = 2x-2
find f(-3)
f(-3) = 2(-3) - 2
f(-3) = -6 -2
f(-3) = -8
After reading a book for English class,100 students were asked whether or not they enjoyed it.
The number of students who like the book is 64 students.
How many students like the book?A fraction is simply a piece of a whole. The number is represented mathematically as a quotient where the numerator and denominator are split.
Given: Total students in English class = 100
Nine twenty-fifths of the class did not like the book.
Thus, the number of students did not like the book = 9/25 × 100
= 36
Therefore, the number of students like the book =Total students- students did not like the book will be:
= 100-36
= 64 students.
The number of students is 64.
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Complete question
After reading a book for English class, 100 students were asked whether or not they enjoyed it. Nine twenty-fifths of the class did not like the book. How many students liked the book?
Find two positive numbers whose difference is 23
23
and whose product is 3408
3408
The two positive numbers are 71 and 48
How to calculate for the two positive numbersWe shall apply algebra in finding the unknown numbers as follows;
Let us represent the two numbers with the letters x and y
so that;
x - y = 23...(1)
xy = 3408...(2)
from equation (1),
x = 23 + y {adding y to both sides}
putting the value 23 + y for x in equation (2) we get;
(23 + y)y = 3408 {by expansion}
23y + y² = 3408
y² + 23y - 3408 = 0 {we rearrange to get a quadratic equation}
we factorize as follows;
y² + 48y - 71y - 3408 = 0
y(y + 48) - 71(y + 48) = 0
(y + 48)(y - 71) = 0
so by factorization; y = -48 or y = 71
the numbers are positive hence we take y = 71
putting the value 71 for y in equation (2) we get;
x71 = 3408 {divide through by 71}
x = 3408/71
x = 48
Thus; 71 - 48 = 23 and 71 × 48 = 3408, and we can say that the two positive numbers are 71 and 23.
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A video streaming company offers two monthly plans. Plan A charges $3 per video viewed plus a flat rate of $8 per month. Plant B charges $5 per video viewed and no additional flat rate.
A write and solve an inequality to determine when the cost of viewing n videos using plan A is less that the cost of viewing n video using plan B _________________
Plan A is less expensive when ____________________
The inequality is 8 + 3 n < 5 n. Plane A is less expensive when n > 4
What is inequality ?
Inequality of wealth in major cities Economic inequality comes in many forms, but two stand out above the rest: wealth inequality as measured by the distribution of wealth and income inequality as measured by the distribution of income.
The idea of inequality, which is the state of not being equal, especially in terms of status, rights, and opportunities1, is at the core of social justice theories. However, because it frequently has diverse meanings to different people, it is prone to misunderstanding in public discourse. However, some distinctions are universal.
Plan A: flat rate $8 per month and $3 per video viewed
Plan B: $5 per video viewed
The number of viewing videos is n
∵ The number of viewing videos is n
→ Plan A
∵ The cost per viewing is $3
∵ Cost per month is $8
∴ The cost = 8 + 3 n
→ Plan B
∵ The cost per viewing is $5
∴ The cost = 5 n
We need the cost of viewing n videos using Plan A is less than the
cost of viewing n videos using Plan B
→ Cost Plan A < Cost Plan B
∵ Cost Plan A is 8 + 3 n
∵ Cost Plan B is 5 n
∴ 8 + 3 n < 5 n
* The inequality is 8 + 3 n < 5 n
∵ 8 + 3 n < 5 n
Subtract 3 n from both sides
∴ 8 < 2 n
Divide both sides by 2
∴ 4 < n
∴ n > 4
* Plane A is less expensive when n > 4
The inequality is 8 + 3 n < 5 n. Plane A is less expensive when n > 4
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What is the difference between the place value of a whole number and a place value of a decimal?
The main differences between the place value of the whole number and the decimal is given below
What is decimal ?
Integer and non-integer numbers are often expressed using the decimal numeral system. The Hindu-Arabic numeral system has been expanded to include non-integer values. Decimal notation is the term used to describe how numbers are represented using the decimal system.
A decimal is a number with a whole and a fractional component. Decimal numbers, which are in between integers, are used to express the numerical value of full and partially whole amounts.
Decimals can be classified as non-recurring (as in not repeating or ending), recurring (as in repeating or not terminating), and decimal fractions (a.k.a. converting a decimal into a fraction whose denominator is a power of ten).
The whole number part of the decimal number is the part that lies strictly to the left of the decimal point, and the place value of each digit in the whole number part is given .
The fractional part of the decimal number is the part that lies strictly to the right of the decimal point.
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help me pleaseee pleaseee pleaseee for a lot of points
Answer:
53 5/7
Step-by-step explanation:
2 + 5/21 = (42 +5)/21 = 47/21
47/21 : 1/12 = x : 2
x = 2 * 47/21 / 1/12
x = 2 * 47/21 * 12
x = 2 * 47/7 * 4
x = 8 * 47/7
x = 376/7
x = 371/7 + 5/7
x = 53 5/7
The length of a rectangle is 4 inches longer than it is wide. If the area is 32 square inches, what are the dimensions of the rectangle?
Answer:
width = 4in
length = 8in
Step-by-step explanation:
A = LxW = 32
L = W + 4
W(W+4) = 32
W² + 4W - 32 = 0
(W + 8)(W - 4) = 0
W ≠ -8 (dimensions cannot be negative)
W = 4in
L = W+4 = 4+4 = 8in
What is the second term in the binomial expansion of (2r – 3s)12?
–73,728r11s
73,728r11s
608,256r10s2
–608,256r10s2
The second term in the binomial expansion of (2r - 3s)¹² is -73728 r¹¹ s option (A) is correct.
What is an expression?It is defined as the combination of constants and variables with mathematical operators.
It is given that:
(2r - 3s)¹²
Using nth term of the binomial expansion is:
= C (12, 2-1) (2r)¹²⁻²⁺¹ (-3s)²⁻¹
After solving
= -73728 r¹¹ s
Thus, the second term in the binomial expansion of (2r - 3s)¹² is -73728 r¹¹ s option (A) is correct.
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The number of men and women receiving bachelor's degrees each year has been steadily increasing. For the years 1970 through the projection of 2014, the number of men receiving degrees (in thousands) is given by the equation y=3.9x+441 , and for women, the equation is y=14.3x+312, where x is the number of years after 1970
The point (12,489) will be the solution of our given equations.
What is Equation?Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.
It demonstrates the equality of the relationship between the expressions printed on the left and right sides. LHS = RHS is a common mathematical formula.
Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.
Given:
number of men given by y=3.9x+441 ,
and for women equation is y=14.3x+312,
Now, Let us find the coordinates of the point of intersection of both lines by equating our both equations as:
3.9x+ 441 = 14.3x + 312
14.3x - 3.9x = 441 - 312
10.4x= 129
x= 129/10.4
x = 12.40
and, y= 3.9(12.4)+ 441
y= 489.36
Hence, the point (12,489) will be the solution of our given equations.
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CAN SOMEONE HELP WITH THIS QUESTION?✨
Answer:
b = 40 , c = 70
Step-by-step explanation:
given a quadratic function in standard form
y = ax² + bx + c ( a ≠ 0 )
then the x- coordinate of the vertex is
x = - [tex]\frac{b}{2a}[/tex]
y = 5x² + bx + c ← is in standard form
with a = 5 and b = b , then
x = - [tex]\frac{b}{10}[/tex]
equating to the given x- coordinate of the vertex , that is - 4
- [tex]\frac{b}{10}[/tex] = - 4 ( multiply both sides by - 10 )
b = 40
then
y = 5x² + 40x + c
substitute (- 4, - 10 ) into the equation and solve for c
- 10 = 5(- 4)² + 40(- 4) + c
- 10 = 5(16) - 160 + c
- 10 = 80 - 160 + c
- 10 = - 80+ c ( add 80 to both sides )
70 = c
then b = 40 and c = 70
Work
Find the radius of a square with perimeter 32cm
Answer: 5.66
Step-by-step explanation:
The perimeter of the square being 32, the sides are 8 units. The diameter of the circumscribing circle is the diagonal of the square which is 8 sec 45 = 11.3 units. Hence, the radius of the circle = 5.66 units.
Given that m_G = 81°, what is m
Answer:
180-81=
Step-by-step explanation:
Find an expression which represents the sum of (7x-y) and (5x+3y) in simplest terms.
An expression that represent the sum of (7x-y) and (5x+3y) in simplest terms is 12x + 2y
How to find the sum of the expressionsThe given expressions are (7x-y) and (5x+3y)
The applying the addition property of equality we have
= (7x - y) + (5x + 3y)
= 7x - y + 5x + 3y
collecting like terms
= 7x + 5x - y + 3y
= 12x + 2y (in simplest terms)
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Aiden runs a farm stand that sells apples and strawberries. Each pound of apples sells
for $2 and each pound of strawberries sells for $3. Aiden made $80 from selling a
total of 35 pounds of apples and strawberries. Write a system of equations that could
be used to determine the number of pounds of apples sold and the number of pounds
of strawberries sold. Define the variables that you use to write the system.
Answer:
Step-by-step explanation:
Dan went for a drive in his new car. He drove at a speed of 64 miles per hour for 102.4 miles. For how many hours did he drive?
___ hours
Answer:
Step-by-step explanation:
Dan drove for 1.6 hours.
64 miles per hour (mph) is the unit rate so divide by 102.4 by 64 and you get 1.6
Equation with one solution 8x-3x+2-x=
[tex]\fbox{4x+2}[/tex]
Simplify:
[tex]8x-3x+2[/tex]
[tex]8x+-3x+2+-x[/tex]
Combine Like Terms:
[tex](8x+-3x+-x)+(2)[/tex]
[tex]4x+2[/tex]
what did the mathematization do to practice over winter break
If Karla has 3 more nickels than quarters and they have a combined value of 135 cents, how many of each coin does she have?
Answer:
Step-by-step explanation:
n=3+d
n -d = 3
5n -5d = 15
5n +10d = 135
5n - 5d = 15
15d = 120
d= 8 dimes
n = 11 nickels
5x11 + 8x10 = 55 + 80= 135
(3×10³) (2× 10²)
Write your answer in scientific notation
Can someone help pls it’s due in about 2 hour
Answer:
see below
Step-by-step explanation:
A
The hypotenuse is ALWAYS the longest side of a right triangle, and located Directly Across from the Right Angle: c = √79
Pythagorean Theorem: a² + b² = c²
a = 2√5
b = ?
c = √79
(2√5)² + b² = (√79)²
20 + b² = 79
b² = 59
√b² = √59
b = √59 = KJ
B
tan=opposite side / adjacent side
From I, KJ is opposite; and IK is adjacent
tan(I) = √59/(2√5) ≈ 1.72
C
cos=adjacent side / hypotenuse
From J, KJ is adjacent; and IJ is the hypotenuse
cos(J) = √59/√79 ≈ 0.86
[tex]5^{2x+1} =25[/tex]
Answer:x=1/2
Step-by-step explanation:
[tex]5^{2x+1} =5^{2}[/tex]
2x+1=2
2x=1
x=1/2