The definition of the inverse function is
[tex]\begin{gathered} f(f^{-1}(x))=x \\ \text{and} \\ f^{-1}(f(y))=y \end{gathered}[/tex]In our case,
[tex]f(x)=2x-1[/tex]Then,
[tex]\begin{gathered} f^{-1}(f(x))=x \\ \Rightarrow f^{-1}(2x-1)=x \\ \Rightarrow f^{-1}(x)=\frac{x+1}{2} \end{gathered}[/tex]We need to verify this result using the other equality as shown below
[tex]\begin{gathered} f^{-1}(x)=\frac{x+1}{2} \\ \Rightarrow f(f^{-1}(x))=f(\frac{x+1}{2})=2(\frac{x+1}{2})-1=x+1-1=x \\ \Rightarrow f(f^{-1}(x))=x \end{gathered}[/tex]Therefore,
[tex]\Rightarrow f^{-1}(x)=\frac{x+1}{2}[/tex]The inverse function is f^-1(x)=(x+1)/2.
We say that a relation is a function if, for x in the domain of f, there is only one value of f(x).
In our case, notice that for any value of x, there is only one value of (x+1)/2=x/2+1/2.
The function is indeed a function, it is a straight line on the plane that is not parallel to the y-axis.
The inverse f^-1(x) is indeed a function
Five companies have each sent one representative to a competition to pitch their newest business idea to a group of angel investors. The conference has 5 time slots designated for one representative to present their company's idea.How many different ways can the representatives be ordered to present their ideas?
Let's use the counting principle:
[tex]\begin{gathered} Let: \\ n=number_{\text{ }}slots=5 \\ T=Total_{\text{ }}number_{\text{ }}of_{\text{ }}ways \end{gathered}[/tex]so:
[tex]T=n\cdot(n-1)\cdot(n-2)...1=n![/tex]So:
[tex]\begin{gathered} T=5! \\ T=120 \end{gathered}[/tex]Answer:
120
The number of different ways can the representatives be ordered to present their ideas is 120 ways.
Given that, 5 companies have each sent one representative to a competition to pitch their newest business idea to a group of angel investors.
What are Permutations?Permutations are different ways of arranging objects in a definite order. It can also be expressed as the rearrangement of items in a linear order of an already ordered set. The symbol nPr is used to denote the number of permutations of n distinct objects, taken r at a time.
Using nPr =n!(n-r)! we get
Here, n=5 and r=5
So, P(n, r) = P(5, 5) = 5!(5-5)!
= 5!/1
= 5×4×3×2×1
= 120 ways
Therefore, the number of different ways can the representatives be ordered to present their ideas is 120 ways.
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Solve for v.37+1=-2v-8v-4
Find an equation of the line through (1,8) and parallel to y = 4x + 8.y=(Type your answer in slope-intercept form.)
First of all, remember that parallel lines are those with equivalent slope. So, the given line is
[tex]y=4x+8[/tex]If the new line we have to find is parallel to this one, that means the slope is
[tex]m=4[/tex]Because the coefficient of x is always the slope.
Now, we know that the new line must pass through (1,8) and it must have a slope of 4. We can use the point-slope formula
[tex]y-y_1=m(x-x_1)[/tex]Replacing the point and the slope, we have
[tex]y-8=4(x-1)[/tex]Then, we solve for y
[tex]y=4x-4+8\rightarrow y=4x+4[/tex]Therefore, the new parallel line is
[tex]y=4x+4[/tex]Image courtesy of NASAWhich of New Zealand's physical features is circled by number 2 on the map above?A. the Northern PeninsulaB. the Southern AlpsC. the Canterbury PlainsD. the Eastern HillsPlease select the best answer from the choices providedABOeCD
C) Canterbury Plains
can you please help me
AB = 3x + 4
BC = 7x + 9
AB + BC = AC
AC = 143
Let us add AB and BC then equate their sum by 143
[tex]AC=AB+BC=3x+4+7x+9=(3x+7x)+(4+9)[/tex]First, step add the like terms
[tex]AC=10x+13[/tex]Equate AC by its length 143
[tex]10x+13=143[/tex]Now we have an equation to solve it
To solve the equation let us move 13 from the left side to the right side by subtracting 13 from both sides
[tex]\begin{gathered} 10x+13-13=143-13 \\ 10x=130 \end{gathered}[/tex]To find x divide both sides by 10 to move 10 from the left side to the right side
[tex]\begin{gathered} \frac{10x}{10}=\frac{130}{10} \\ x=13 \end{gathered}[/tex]Now let us find AB and BC
Substitute x by 13 in each expression
AB = 3(13) + 4 = 39 + 4 = 43
BC = 7(13) + 9 = 91 + 9 = 100
The length of AB is 43 units
The length of BC is 100 units
Find the LCD of the list of fractions. 11/20, 1/18, 13/90
LCD state for Least Common Denominator
The given fraction are :
[tex]\frac{11}{20},\text{ }\frac{1}{18},\text{ }\frac{13}{90}[/tex]For the least common denominator, first find the LCM of all the denominator of the given values:
Denominator are : ( 20, 18, 90)
LCM of (20,18, 90) = 180
So, the fraction will value can be written as :
[tex]\begin{gathered} \frac{11}{20}\text{ to make denominator equal to 180,} \\ \text{ Multiply up \& down by 9} \\ \frac{11\times9}{20\times9}=\frac{99}{180} \\ \text{ } \\ \frac{1}{18}\text{ to make denominator equal to 180} \\ \text{ Multiply up and down by 10} \\ \frac{1\times10}{18\times10}=\frac{10}{180} \\ \\ \frac{13}{90},\text{ to make denominator equal to 180} \\ \text{Multiply up and down by 2} \\ \frac{13\times2}{90\times2}=\frac{26}{180} \end{gathered}[/tex]Thus, the fraction will convert as :
[tex]\begin{gathered} \frac{11}{20}=\frac{99}{180} \\ \frac{1}{18}=\frac{10}{180} \\ \frac{13}{90}=\frac{26}{180} \end{gathered}[/tex]The least common denominator is 180
Answer : LCD of 11/20, 1/18, 13/90 is 180
Question 8: What is the measure of Angle C?*c525°47°43°1330
SOLUTION
Angle C is 133 degrees
From the image , angle c is the same as angle a, reason been that they are vertically opposite angles and they are always equal.. let us call angle c and a = x
Angle b = 47 degrees, because they are both vertically opposite angles, and they are always equal.
Angle c + angle a + angle b + 47 = 360 ( sum of angles at a point)
x + x + 47 + 47 = 360
2x + 94 =360
2x = 360-94
2x =266
x= 266/2
x=133 degrees
So angle C is 133 degrees
Option D
|x-2|-3 >or equal to 2
By solving the linear inequation it is obtained that [tex]x \leq -3[/tex] or [tex]x \geq 7[/tex].
What is linear inequation?
Expressions with linear inequalities compare any two values using inequality symbols like ‘<’, ‘>’, ‘≤’ or ‘≥’. These values could be either numerical, algebraic, or both. Examples of numerical inequalities include 1011 and 20>17, while algebraic inequalities include x>y, y19-x, and x z > 11 (also called literal inequalities). Here is a lesson on linear inequalities for class 11. Inequality in mathematics, linear inequalities, graphing of linear inequalities, as well as detailed examples are all covered in this article.
Here,
The given linear inequation is
[tex]|x - 2| - 3 \geq 2[/tex]
Now,
[tex]|x -2| - 3 \geq 2\\|x - 2| \geq 2+3\\|x-2| \geq 5\\[/tex]
For [tex]x \geq 2\\[/tex]
[tex]x - 2 \geq 5\\x \geq 2 + 5\\x\geq 7[/tex]
For [tex]x < 2[/tex]
[tex]2 - x \geq 5\\x \leq 2 - 5\\x \leq -3[/tex]
So the solution set is [tex]x \leq -3[/tex] or [tex]x \geq 7[/tex]
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In the rectangle below, B D = 4x – 2, AC = 5x-11, and m ZAED = 82º.Find AE and m ZECB.BEAE =m ZECB =DС
Given :
[tex]\begin{gathered} BD\text{ = 4x + 2} \\ AC\text{ = 5x - 11} \\ \angle AED=82^0 \end{gathered}[/tex]Required :
[tex]AE\text{ , }\angle\text{ ECB}[/tex]Recall from the properties of a rectangle that
[tex]\text{The diagonals have the same length}[/tex]Hence :
[tex]\begin{gathered} AC\text{ = BD} \\ 5x\text{ - 11 = 4x -2 } \\ \text{collect like terms} \\ 5x\text{ - 4x = 11 - 2} \\ x\text{ = 9} \end{gathered}[/tex]express in scientific notation (9.3 x 10^7) ÷ 23,000 = ? (round to the nearest tenth.)
Given:
[tex]\frac{9.3\times10^7}{23000}[/tex]Let's perform the division and express the quotient in scientific notation.
We have:
[tex]\frac{9.3\times10^7}{23000}=\frac{9.3\times10000000}{23000}=\frac{93000000}{23000}=4043.478261[/tex]Express 4043.478261 in scientific notation:
[tex]undefined[/tex]which of the following is the correct factorization of the polynomial below?27x^3+1000
The polynomial is given to be:
[tex]27x^3+1000[/tex]We can rewrite this expression by applying the knowledge of exponents:
[tex]\Rightarrow(3x)^3+10^3[/tex]Apply the sum of cubes formula:
[tex]x^3+y^3=\left(x+y\right)\left(x^2-xy+y^2\right)[/tex]Therefore, we have:
[tex]\left(3x\right)^3+10^3=\left(3x+10\right)\left(3^2x^2-10\cdot \:3x+10^2\right)[/tex]Hence, we can simplify the expression to give the answer:
[tex]27x^3+1000=\left(3x+10\right)\left(9x^2-30x+100\right)[/tex]The correct option is OPTION B.
find the odds of an event occurring given the probability of the event 6/7
Odds is the ratio of favourable outcomes to non-favourable outcomes:
Let:
P = probability of an event occurring = 6/7
Q = probability of the event not occurring = 1 - P = 1 - 6/7 = 1/7
Therefore, the odds will be:
[tex]\frac{P}{Q}=\frac{\frac{6}{7}}{\frac{1}{7}}=6[/tex]Find the 8th term of the sequence using the explicit formula: 2 × (7)(n - 1).1647086164706816478061647860
Given:
The explicit formula is
[tex]2\times7^{(n-1)}[/tex]Required:
To find the 8th term of the sequence.
Explanation:
For n=8,
[tex]\begin{gathered} =2\times7^{(8-1)} \\ \\ =2\times7^7 \\ \\ =2\times7\times7\times7\times7\times7\times7\times7 \\ \\ =1647086 \end{gathered}[/tex]Final Answer:
The first option is correct.
[tex]1647086[/tex]When 3.5 is added to 7 times a number the result is 65.1 find the number
Let x be the number we are looking for; therefore, 7 times a number is '7x'.
Then, 3.5 added to 7 times a number is
[tex]3.5+7x[/tex]Thus, the whole equation is
[tex]\begin{gathered} 3.5+7x=65.1 \\ \Rightarrow7x=65.1-3.5=61.6 \\ \Rightarrow x=\frac{61.6}{7}=8.8 \end{gathered}[/tex]Hence, the number is 8.8
The parent tangent function is horizontally compressed by a factor of 1/2 and reflected over the x-axis. Which equation could represent function g.the result of this transformation?OA. g(x) = -tan(2x)O B. g(x) = tan(-1/2x)OC. g(x) = tan(-2x)OD. g(x) = -tan(1/2x)
Given :
The parent tangent function is horizontally compressed by a factor of 1/2 and reflected over the x-axis.
Explanation :
To find the equation of g.
The tangent function is
[tex]f(x)=\tan x[/tex]It is horizontally compressed by a factor of 1/2.
Then the function becomes
[tex]g(x)=\tan (\frac{1}{2}x)[/tex]After that it is reflected over x-axis.
[tex]g(x)=-\tan (\frac{1}{2}x)[/tex]Answer :
Hence the result of the transformation is
[tex]g(x)=-\tan (\frac{1}{2}x)[/tex]The correct option is D.
which is equal to 73.5÷by 15
The answer to this division is 4.9.
You can also multiply the numerator (dividend) and the denominator (divisor) by 10, so you can have the equivalent division:
[tex]\frac{73.5}{15}\cdot\frac{10}{10}=\frac{735}{150}=4.9[/tex]And proceed as before. The result will be the same.
Use a system of equations to solve the following problem.The sum of three integers is 244. The sum of the first and second integers exceeds the third by 48. The third integer is 36 less than the first Findthe three integersAnswer How to enter your answer topens in new windon 5 PointsKeypadKeyboard Shartofirst integer =second integer =third integer =
Given:
The sum of three integers is 244. The sum of the first and second integers exceeds the third by 48. The third integer is 36 less than the first.
Aim:
We need to find the values of all three integers.
Explanation:
Let x be the first integer.
Let y be the second integer.
Let z be the third interger.
The sum of three integers is 244.
[tex]x+y+z=244[/tex]The sum of the first and second integers exceeds the third by 48.
[tex]x+y=z+48[/tex]The third integer is 36 less than the first.
[tex]z=x-36[/tex]Substitute z=x-36 in the equation x+y=z-48 .
[tex]x+y=x-36+48[/tex][tex]x+y=x+12[/tex]Subtract x from both sides of the equation.
[tex]x+y-x=x+12-x[/tex][tex]y=12[/tex]Substitute z=x-36 and y=12 in the equation x+y+z=244.
[tex]x+12+x-36=244[/tex]Add 24 to both sides of the equation.
[tex]2x-24+24=244+24[/tex][tex]2x=268[/tex]Divide both sides by 2.
[tex]\frac{2x}{2}=\frac{268}{2}[/tex][tex]x=134[/tex]Substitute x=134 in the equation z=x-36
[tex]z=134-36[/tex][tex]z=98[/tex]We get x=128, y=12 and z =98.
Final answer:
first integer = 128
second integer =`12
third integer = 98.
the function is shaped like a u what is the standard form or basic function.
The function shaped like a U is in the form of a basic quadratic equation and is represented as a parabola.
The graph of a quadratic function is a U-shaped curve called a parabola. An important feature of graphs is that they have extreme points called vertices.
When the parabola opens upwards, the vertex represents the lowest point of the graph, or the minimum value of the quadratic function. When the parabola opens downwards, the vertex represents the highest point or maximum of the graph.
In both cases, the vertex is the inflection point of the graph. Graphs are also symmetrical about a vertical line through the vertices called the axis of symmetry.
The standard form or basic function for a parabola will be in the form of a quadratic function such as -
[tex]f(x)=a(x-h)^{2} +k[/tex]
where, [tex](h,k)[/tex] = vertex
Thus, the function shaped like a U is in the form of a basic quadratic equation and is represented as a parabola.
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Consider the following linear equation.2y = -1-ainStep 2 of 2: Graph the line.
As given by the question
There are given that the equation
[tex]y=-1-\frac{2}{5}x[/tex]Now,
The graph of the line is given below:
Simplify 310x + 16y + 310x + 56y ( i need help)
Answer:
[tex]620x+72y[/tex]
Step-by-step explanation:
[tex]310x+16y+310x+56y \\ \\ =310x+310x+16y+56y \\ \\ =620x+72y[/tex]
.1.2_Updated_FY21 Question: 1-3 The elevation of the Vander's home is -108 feet. The elevation of the Gail's home is exactly of that depth below sea level. What is the elevation of the Gail's home in feet? -36 -72 -162 -180
Given:
Elevation of Vander's home = -108 feet
Elevation of Gali's home is ⅔ of that depth below sea level.
Thus, the elevation of Gali's home is:
⅔ of -108 feet =
[tex]\frac{2}{3}(-108)\text{ = }\frac{2(-108)}{3}=\frac{-216}{3}=\text{ -72 f}eet[/tex]We know that the elevation of Vander's home is already below sea level since it's a negative value.
Therefore, since the elevation of Gali's home is ⅔ of the depth of Vander's home below sea level, the elevation of Gali's home is:
-72 feet
ANSWER:
-72 feet
Could I please get help with this. I can’t seem to figure out the answers to each of the figures after multiple tries.
Explanation:
Two figures are congruent when they have the same size and shape and two figures are similar when they have the same shape but not necessarily the same size. In similar figures, the ratio of the corresponding sides is constant.
Answer:
Then, for each pair, we get:
Let f(x) = 9 - x, g (x) = x*2 + 2x - 8, and h (x) = x - 4
Solution
We are given the following functions
[tex]\begin{gathered} f(x)=9-x \\ g(x)=x^2+2x-8 \\ h(x)=x-4 \end{gathered}[/tex]g(x) + f(x)
[tex]\begin{gathered} g(x)+f(x)=(x^2+2x-8)+(9-x) \\ \\ g(x)+f(x)=x^2+2x-8+9-x \\ \\ g(x)+f(x)=x^2+x+1 \end{gathered}[/tex]h(x) - f(x)
[tex]\begin{gathered} h(x)-f(x)=(x-4)-(9-x) \\ \\ h(x)-f(x)=x-4-9+x \\ \\ h(x)-f(x)=2x-13 \end{gathered}[/tex]f o h(10)
[tex]\begin{gathered} First \\ h(x)=x-4 \\ h(10)=10-4 \\ h(10)=6 \\ and \\ f(x)=9-x \\ f(6)=9-6 \\ f(6)=3 \\ Now,\text{ to solve} \\ foh(10)=f(h(10)) \\ foh(10)=f(6) \\ \\ foh(10)=3 \end{gathered}[/tex]3 * g(-1)
[tex]\begin{gathered} First, \\ g(x)=x^2+2x-8 \\ g(-1)=(-1)^2+2(-1)-8 \\ \\ g(-1)=1-2-8 \\ \\ g(-1)=-9 \\ Now\text{ to solve} \\ 3g(-1)=3\times g(-1) \\ \\ 3g(-1)=3\times-9 \\ \\ 3g(-1)=-27 \end{gathered}[/tex]h(x) * h(x)
[tex]\begin{gathered} h(x)=x-4 \\ Now, \\ h(x)*h(x)=(x-4)(x-4) \\ \\ h(x)*h(x)=x^2-8x+16 \end{gathered}[/tex]g(x)/h(x)
[tex]\frac{g(x)}{h(x)}=\frac{x^2+2x-8}{x-4},\text{ }x\ne4[/tex]I'm having trouble finding the length of NP and MN, thinking it has something to do with tan, cos, and sin, but not completely sure.
Bisects: to divide into two equal parts.
In this case, DB is bisecting the ∠ABC, then the ∠ABD
As OP is bisecting ∠MON, that means that ∠NOP and ∠POM have the same measure.
Then:
[tex]m\angle MON=m\angle NOP+m\angle POM[/tex]As ∠NOP = ∠POM, we get:
[tex]m\angle MON=m\angle NOP+m\angle NOP=2\cdot m\angle NOP[/tex]Replacing the value we get:
[tex]m\angle MON=2\cdot20=40[/tex]Based on this, we can use the trigonometric functions, as we have an angle and one side. Specifically, the tangent function:
[tex]\tan \alpha=\frac{opposite}{\text{adyacent}}[/tex]First, to calculate NP, we get the following:
[tex]\tan 20=\frac{NP}{6}[/tex]Isolating for NP:
[tex]NP=6\cdot\tan 20[/tex][tex]NP=2.18[/tex]Then, calculating for MN we get the following:
[tex]\tan 40=\frac{MN}{6}[/tex]Isolating for MN:
[tex]MN=6\cdot\tan 40[/tex][tex]MN=5.03[/tex]Answer:
• NP = 2.18
,• MN = 5.03
a line has a slope of 3 and a y-i yet dot of 5. what is it’s equation in slope-intercept form? write you answer using integers, proper fractions, and improper fractions in simplest form.
y = 3x + 5
Explanation:slope = 3
y - intercept = 5
To get the equation in slope intercept form, we'll use:
[tex]\begin{gathered} y\text{ = mx + b} \\ m\text{ = slope} \\ b\text{ = y-intercept} \end{gathered}[/tex][tex]\begin{gathered} \text{The equation becomes:} \\ y\text{ = 3x + 5} \\ \end{gathered}[/tex]Find the equation of the linear function represented by the table below in slope-intercept form.X1234y691215******
Given:
Given a table.
Required:
To find the equation of the linear function.
Explanation:
From the table
[tex]\begin{gathered} (x1,y1)=(1,6) \\ (x2,y2)=(2,9) \end{gathered}[/tex]The general form of equation is
[tex]y=mx+b[/tex]Here the slope is
[tex]\begin{gathered} m=\frac{y2-y1}{x2-x2} \\ \\ =\frac{9-6}{2-1} \\ \\ =\frac{3}{1} \\ \\ =3 \end{gathered}[/tex]So
[tex]y=3x+b[/tex]Now we have to find the value of b, by using the point (1,6)
[tex]\begin{gathered} 6=3(1)+b \\ \\ 6-3=b \\ \\ b=3 \end{gathered}[/tex]Now
[tex]y=3x+3[/tex]Final Answer:
The linear equation is
[tex]y=3x+3[/tex]Given the system of equations: 8x + 14y = 4 and -6x - 7y = - 10, what would youmultiply the bottom equation by to eliminate y when adding the two equationstogether?
We need to multiply the second equation by 2 to eliminate y when adding the two equations
Help me with math and explain it in a short solution
The perimeter is the sum of all the sides of a geometric figure. Since it is a parallelogram, then its opposite sides are equal, so
[tex]\begin{gathered} QR=TS \\ \text{and} \\ QT=RS \end{gathered}[/tex]In the graph, we can see that the distance between points Q and R is 7 units. To find the distance between points Q and T we can use the formula of the distance between two points in the plane, that is,
[tex]\begin{gathered} d=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2} \\ \text{ Where} \\ (x_1,y_1)\text{ and }(x_2,y_2)\text{ are the coordinates of the points } \end{gathered}[/tex]Then, we have
[tex]\begin{gathered} Q(-3,3) \\ T(-5,-3) \\ d=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2} \\ \text{ Replace} \\ d=\sqrt[]{(-5-(-3))^2+(-3-3)^2} \\ d=\sqrt[]{(-5+3)^2+(-3-3)^2} \\ d=\sqrt[]{(-2)^2+(-6)^2} \\ d=\sqrt[]{4+36} \\ d=\sqrt[]{40} \end{gathered}[/tex]Finally, we have
[tex]\begin{gathered} \text{ Perimeter }=QR+RS+TS+QT \\ \text{ Perimeter }=7+\sqrt[]{40}+7+\sqrt[]{40} \\ \text{ Perimeter }=26.65 \end{gathered}[/tex]Therefore, the perimeter of parallelogram QRST is 26.65 units and the correct answer is option B.
A single die is rolled 4 times. Find the probability of getting at least one 6.
When a dice is rolled the probability of getting one 6 is,
[tex]P(\text{Getting one 6) = }\frac{1}{6}[/tex]The probability of not getting 6 when a dice is rolled is ,
[tex]P(\text{Not getting 6) = }\frac{5}{6}[/tex]The probability of getting 6 is independent on how many times the dice is rolled.
The probability of not getting 6 is given as,
[tex]P(\text{ getting atleast one 6) = 1 - P(Not getting 6)}[/tex]Therefore the probability of getting atleast one 6 when a dice is rolled 4 times is calculated as,
[tex]\begin{gathered} P(\text{Getting 6) = 1 - (}\frac{5}{6})^4 \\ P(\text{Getting 6) = 1 - }\frac{625}{1296} \\ P(\text{Getting 6) = }0.5177 \\ \end{gathered}[/tex]Thus the probability of getting atleast one 6 when a dice is rolled 4 times is 0.5177 .
Allison stated that 48/90 is a terminating decimal equal to 0.53. Why is she true or why is she wrong.
Answer:
She was Wrong, because it is not a terminating decimal
Explanation:
Given the fraction;
[tex]\frac{48}{90}[/tex]Let us reduce the fraction to its least form, then convert it to decimal.
[tex]\frac{48}{90}=\frac{8}{15}[/tex]converting to decimal we have;
The decimal form of the given fraction is;
[tex]\begin{gathered} 0.533\ldots \\ =0.53\ldots \end{gathered}[/tex]Which is not a terminating decimal, because it has an unending, repeatitive decimal.
Therefore, she was Wrong, because it is not a terminating decimal