First, find the inverse function of f(x). To do so, replace y=f(x) and isolate x from the equation:
[tex]\begin{gathered} f(x)=\frac{(x+2)}{3} \\ \Rightarrow y=\frac{(x+2)}{3} \\ \Rightarrow3y=x+2 \\ \Rightarrow3y-2=x \\ \therefore x=3y-2 \end{gathered}[/tex]Swap x and y to find the inverse function:
[tex]y=3x-2[/tex]Replace y=f¨-1(x):
[tex]f^{-1}(x)=3x-2[/tex]Notice that the inverse function is written in slope-intercept form, and the coefficient of the variable x is the slope. In this case, the coefficient of x is 3.
Therefore, the slope of the inverse function of f(x) is 3.
the cost y in cents of an ounce X of almonds is represented by the equation y equals 50 x the graph shows the cost of peanuts find the. cost per ounce gor each type of nut which is more expensive
From the equation for the cost of almonds (y=50x) we see that the cost of almonds per ounce is: 50 cents
Given that two ounce of peanuts cost 50 cents, we deduce that 1 ounce of peanuts cost: 50/2 cents, 25 cents
In conclusion, almonds are more expensive per ounce.
Emmanuel invests $3600 and Kelsey invests $2400. Both investments eam 3.8%annual interest. How much longer will it take Kelsey's investment to reach $10,000than Emmanuel's investment?
EXPLANATION
We can build the following equations in order to represent the Emmanuel earnings:
[tex]y_1=3600(1+0.038)^x[/tex][tex]y_2=10000[/tex]Matching both expressions:
[tex]10000=3,600(1+0.038)^x[/tex][tex]\mathrm{Switch\: sides}[/tex][tex]3600\mleft(1+0.038\mright)^x=10000[/tex][tex]\mathrm{Divide\: both\: sides\: by\: }3600[/tex][tex]\frac{3600\left(1+0.038\right)^x}{3600}=\frac{10000}{3600}[/tex][tex]\mathrm{Simplify}[/tex][tex]\mleft(1+0.038\mright)^x=\frac{25}{9}[/tex][tex]\mathrm{If\: }f\mleft(x\mright)=g\mleft(x\mright)\mathrm{,\: then\: }\ln \mleft(f\mleft(x\mright)\mright)=\ln \mleft(g\mleft(x\mright)\mright)[/tex][tex]\ln \mleft(\mleft(1+0.038\mright)^x\mright)=\ln \mleft(\frac{25}{9}\mright)[/tex][tex]\ln \mleft(\mleft(1+0.038\mright)^x\mright)=x\ln \mleft(1+0.038\mright)[/tex][tex]x\ln \mleft(1+0.038\mright)=\ln \mleft(\frac{25}{9}\mright)[/tex][tex]\mathrm{Divide\: both\: sides\: by\: }\ln \mleft(1.038\mright)[/tex][tex]\frac{x\ln\left(1+0.038\right)}{\ln\left(1.038\right)}=\frac{\ln\left(\frac{25}{9}\right)}{\ln\left(1.038\right)}[/tex]Simplify:
[tex]x=\frac{\ln\left(\frac{25}{9}\right)}{\ln\left(1.038\right)}[/tex]In decimal form, this is equivalent to 27.39 years.
Now, applying the same reasoning to the Kelsey investment:
[tex]y_1=2400(1+0.038)^x[/tex][tex]y_2=10000[/tex]Matching both expressions:
[tex]10000=2400(1+0.038)^x[/tex][tex]Switch\: sides[/tex][tex]2400\mleft(1+0.038\mright)^x=10000[/tex][tex]\mathrm{Divide\: both\: sides\: by\: }2400[/tex][tex]\frac{2400\left(1+0.038\right)^x}{2400}=\frac{10000}{2400}[/tex][tex]\mathrm{Simplify}[/tex][tex]\mleft(1+0.038\mright)^x=\frac{25}{6}[/tex]Applying the exponent rule:
[tex]\mathrm{If\: }f\mleft(x\mright)=g\mleft(x\mright)\mathrm{,\: then\: }\ln \mleft(f\mleft(x\mright)\mright)=\ln \mleft(g\mleft(x\mright)\mright)[/tex][tex]\ln \mleft(\mleft(1+0.038\mright)^x\mright)=\ln \mleft(\frac{25}{6}\mright)[/tex]Apply log rule:
[tex]\ln \mleft(\mleft(1+0.038\mright)^x\mright)=x\ln \mleft(1+0.038\mright)[/tex][tex]x\ln \mleft(1+0.038\mright)=\ln \mleft(\frac{25}{6}\mright)[/tex][tex]\mathrm{Divide\: both\: sides\: by\: }\ln \mleft(1.038\mright)[/tex][tex]\frac{x\ln\left(1+0.038\right)}{\ln\left(1.038\right)}=\frac{\ln\left(\frac{25}{6}\right)}{\ln\left(1.038\right)}[/tex]Simplify:
[tex]x=\frac{\ln\left(\frac{25}{6}\right)}{\ln\left(1.038\right)}[/tex]Expressing in decimal form:
[tex]x=38.26[/tex]This is equivalent to 38.26 years to reach 10000 to the Kesley investment.
Comparing both investments:
Kesley Investment - Emmanuel Investment =
= 38.26 - 27.39 = 10.87
In conclusion, It will take 10.87 more years for Kelsey to reach $10,000
the vikings scored more points than packers in a ratio of 7:2. if the packers scored 14 points, how many points did the viking score
The ratio means that for every 7 points the vikings scored the packers only scored 2. Therefore we can represent this situation as a equation with fractions as shown below:
[tex]\text{ v i kings = }\frac{7}{2}\cdot packers[/tex]If the packers scored a total of 14 points, then the vikings scored:
[tex]\begin{gathered} v\text{ i kings = }\frac{7}{2}\cdot14 \\ v\text{ i kings = 7}\cdot7\text{ = 49} \end{gathered}[/tex]The vikings scored 49 points.
If you are subtracting (3x2 + x) - (-2x2 + x), which of the following choices correctly adds the opposite of the second polynomial? (1 point) A A (3.x² + x) - (2x - x) B (3.rº + r) + (2x2 + x) C (3x2 + x) + (-2x2 – x)D (3.2? + x) + (2x2 – x) .
The given expression:
[tex](3x^2+x)-(-2x^2+x)[/tex]Open the brackets:
[tex]\begin{gathered} (3x^2+x)-(-2x^2+x) \\ (3x^2+x)-(-2x^2+x)=(3x^2+x)+2x^2-x \\ (3x^2+x)-(-2x^2+x)=(3x^2+x)+(2x^2-x) \end{gathered}[/tex]Answer: D)
[tex](3x^2+x)+(2x^2-x)[/tex]
Simplify (2 + 6)2 + (1 - 2)
We have the next expression
(2+6)2+(1-2)
First we do the multiplication
(
The length of a rectangle is 4 feet longer than twice the width. if the perimeter is 86 feet find the length and width of the rectangle
Let w be the width of the rectangle.
Let l be the length of the rectangle.
But l = w+4
Also,
[tex]\begin{gathered} Perimeter=2\mleft(l+w\mright) \\ 86=2((w+4)+w) \\ 86=2(2w+4) \\ 86=4w+8 \\ \text{Subtract 8 from both sides} \\ 86-8=4w+8-8 \\ 4w=78 \\ \text{Divide both sides by 4 to get} \\ \frac{4w}{4}=\frac{78}{4} \\ w=19.5ft \end{gathered}[/tex]Width = 19.5ft
Length = 19.5 + 4 = 23.5ft
19. In a small town named Danville, the population today is 6999. For the last sixyears, the population has grown by 20% every year. If the growth stay thesame, what is the expected population in Danville 4 years from today?O 14487O14496O14560OAll changes saveO 14513
From the statement, we know that the population in Danville:
• today is P₀ = 6999,
,• for the last six years, the population has grown by r = 20% = 0.2 every year.
Supposing the same growth rate for the following years, we must compute the expected population in Danville n = 4 years from today.
(1) The general formula for the population with constant growth is given:
[tex]P_n=P_0\cdot(1+r)^n.[/tex]Where:
• P₀ = 6999 is the initial population,
,• r = 20% = 0.2 is the growth rate in decimals,
,• n = the number of years from today.
(2) Replacing the data from above, we have the following formula:
[tex]P_n=6999\cdot(1+0.2)^n=6999\cdot1.2^n.[/tex](3) Evaluating this formula for n = 4, we get:
[tex]P_4=6999\cdot1.2^4\cong14513.[/tex]So the expected population in Danville 4 years from today is about 14513 people.
Answer14513
Using the given graphWhat is the values of x when f(x)= -6
Given
The graph,
To find:
The value of x when f(x)=-6.
Explanation:
It is given that,
From, the above graph it is clear that, at y = -6, x=5 and x=9.
Since f(x)=y.
Then f(x)=-6 at x=5 and x=9.
Hence, the values of x is 5 and 9.
Joshua decides to use synthetic division to find the the quotient of (equation provided above).What value from the divisor should he write in front of the coefficients?-332-2
Firstly, we need to understand what is meant by synthetic division.
In synthetic division, we write out coefficients of the divisor.
The coefficients of the divisor are;
3 , 6 , 2 , 9 , 10
Now, lets take a look at the term which we would be dividing with. We simply set the term which is x + 2 to 0
This gives x + 2 = 0
and thus, x = -2
This is the value w
Write an expression for the perimeter of the rectangle below.(Perimeter is the distance around a figure)
As you say well, the perimeter (P) of a figure is the "distance around it". What we need is just to add up to the length of all sides of the rectangle. Doing that, in this case, leads us to
[tex]P=n+(n+2)+n+(n+2)[/tex][tex]P=4n+4[/tex]If you want a more elegant way to write it, it would be
[tex]P=4(n+1)[/tex]Find the equations of any horizontal asymptotes. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)
ANSWER
OPTION B
The horizontal asymptote is 7
STEP-BY-STEP EXPLANATION
Given information
[tex]f(x)\text{ = }\frac{7|x|}{x\text{ + 2}}[/tex]Therefore, the correct option is B
Part B
The horizontal asymptote of a rational function can be determined by looking at the degree of numerator and denominator.
Therefore, the horizontal asymptote is 7
Plot the midpoint of segment AB . Then give its coordinates. I need help on number 1. Thank you
To find the midpoint of the segment AB.
The point A is,
[tex]A(-3,3)[/tex]The point B is,
[tex]B(5,-2)[/tex]So to find the midpoint value. we use the formula,
[tex]\text{Midpoint of AB=(}\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})[/tex]From the point A and B . we have the values,
[tex]x_1=-3,y_1=3,x_2=5,y_2=-2[/tex]Subsituting the value we get,
[tex]\begin{gathered} M\text{idpoint of AB=(}\frac{-3+5}{2},\frac{3+(-2)}{2}) \\ \text{ =(}\frac{2}{2},\frac{1}{2}) \\ \text{ =(1,}\frac{1}{2}) \end{gathered}[/tex]So the midpoint of AB is,
[tex](1,\frac{1}{2})[/tex]i need help question
Given:
[tex]4x^3-18x^2-48x-3[/tex]find: critical number of point of a function.
Explanation: critical numbers are values of a function where the function's tangent lines.
[tex]\begin{gathered} y=4x^3-18x^2-48x-3 \\ \frac{dy}{dx}=12x^2-36x-48=0 \\ x^2-3x-4=0 \\ x^2-(4-1)x-4=0 \\ x^2-4x+x-4=0 \\ x(x-4)+(x-4)=0 \\ (x-4)(x+1)=0 \\ x=4,-1 \end{gathered}[/tex]Final answer: there are two critical number x=4 and -1.
(Simplify your answer. Use integers or fractions for any numbers in the expression.)
Given:
[tex]f(x)=4x+2[/tex][tex]y=4x+2[/tex][tex]x=4y+2[/tex][tex]4y=x-2[/tex][tex]y=\frac{x-2}{4}[/tex][tex]f^{-1}(x)=\frac{x-2}{4}[/tex]b)
Domain of f : Domain is the set of all real numbers.
c)
Let Red,blue and green lines represent the graph of f, f inverse and y=x
Evaluate each expression if f = -6, g = 7, and h=9. solve10. g -7 11.-h-(-9) 12. f- g
hello
to solve this question, we simply need to substitute the values of the variables into the expression
[tex]\begin{gathered} f=-6 \\ g=7 \\ h=9 \end{gathered}[/tex][tex]\begin{gathered} 10. \\ g-7 \\ g=7 \\ 7-7=0 \end{gathered}[/tex]number 10 = 0
[tex]\begin{gathered} 11. \\ -h-(-9) \\ h=9 \\ 11.-9-(-9)=-9+9=0 \end{gathered}[/tex]number 11 = 0
[tex]\begin{gathered} 12. \\ f-g \\ f=-6 \\ g=7 \\ f-g=-6-7=-13 \end{gathered}[/tex]number 12 = -13
Two sides of an isosceles triangle have lengths of 4 and 8. What is the length of the third side?A. 4B. 6C. 8D. 12
SOLUTION:
An isosceles triangle has two sides equal. Thus, this isosceles triangle would either have for a third side 4 or 8.
Using the triangle inequality, however, we can rule out 4. This is because;
[tex]4+4=8[/tex]Therefore, we have no triangle;
Therefore, the answer is;
[tex]8[/tex]4. A cell phone tower is anchored by 2 cables for support. They stretch from the top of the tower to the ground. The angle of depression from the top of the tower to the point at which the cable reaches the ground is 23°, and distance from the bottom of the tower to the point at which the cable reaches the ground is 60 feet. How long is the cable?
Let's begin by identifying key information given to us:
We are given one known angle & one known side
[tex]\theta=23^{\circ},adjacent=60ft,hypotenuse=\text{?}[/tex]We will solve using Trigonometric Ratio (SOHCAHTOA) & in this case, CAH:
[tex]\begin{gathered} CAH\Rightarrow cos\theta=\frac{adjacent}{hypotenuse} \\ cos\theta=\frac{adjacent}{hypotenuse} \\ cos23^{\circ}=\frac{60}{hypotenuse} \\ hypotenuse\cdot cos23^{\circ}=60 \\ hypotenuse=\frac{60}{cos23^{\circ}}=\frac{60}{0.9205} \\ hypotenuse=65.18\approx65 \\ hypotenuse=65ft \end{gathered}[/tex]Therefore, the cable is 65 feet long
how do I solve and what would be the answer
Given
The area of the rectangle is represented as,
[tex]A=14x^3-35x^2+42x[/tex]And, the length is represented as,
[tex]l=7x[/tex]To find the breadth of the rectangle.
Now,
The area of the rectangle is given by,
[tex]A=l\times b[/tex]Then,
[tex]b=\frac{A}{l}[/tex]Substitute the values of A and l in the above equation.
Then,
[tex]b=\frac{14x^3-35x^2+42x}{7x}[/tex]Since the common terrm in 14x^3-35x^2+42x is 7x.
Then,
[tex]\begin{gathered} b=\frac{7x(2x^2-5x+6)}{7x} \\ b=2x^2-5x+6 \end{gathered}[/tex]Hence, the breadth of the rectangle is 2x^2-5x+6.
Suppose that there are two types of tickets to a show: advance and same-day. The combined cost of one advance ticketand one same-day ticket is $65. For one performance, 25 advance tickets and 30 same-day tickets were sold. The totalamount paid for the tickets was $1750. What was the price of each kind of ticket?Advance ticket: s[]Same-day ticket: s[]
Let's call a the price of advance tickets, and s the price of same-day tickets.
The combined price of one and one is $65. Then, we can establish the following equation:
[tex]a+s=65[/tex]When 25 advance tickets and 30 same-day tickets were sold, the total amount paid for the tickets was $1750. Then, the sum of the total amount of money paid for advanced tickets (25a, the number of tickets paid multiplied by their price), and the total amount of money paid for the same-day tickets (30s, following the same logic), should add up to $1750. Representing that in an equation:
[tex]25a+30s=1750[/tex]Now we have a system of two equations and two unknowns (the price for each kind of ticket).
We can solve the system by using the substitution method.
We can isolate one of the variables from the first equation, and then replace the expression on the second equation. Solving for s in the first equation:
[tex]s=65-a[/tex]Now, we can replace that expression in the second equation instead of s:
[tex]25a+30\cdot(65-a)=1750[/tex]Now we have one equation with only one unknown. We can easily solve now the value of a:
[tex]\begin{gathered} 25a+30\cdot65-30a=1750 \\ 25a-30a=1750-30\cdot65 \\ -5a=1750-1950 \\ a=\frac{-200}{-5} \\ a=40 \end{gathered}[/tex]Then, the price of the advance ticket is $40.
Recalling that the combined price of one ticket of each class is $65:
[tex]a+s=65[/tex]Replacing the price of the advance ticket and solving:
[tex]\begin{gathered} 40+s=65 \\ s=65-40 \\ s=25 \end{gathered}[/tex]The price of the same-day tickets is $25
Solve the equation 1/4 (x+2)+5=-x
We have to solve the equation
[tex]-\frac{1}{4}(x+2)+5=-x[/tex]So we are going to let the variable x in one side of the equation and the constants (the numbers) in the other, but fisrt we open the parentheses
[tex]\begin{gathered} -\frac{1}{4}x-\frac{2}{4}+5=-x \\ \\ -\frac{1}{4}x-\frac{1}{2}+5=-x \end{gathered}[/tex]Now we can solve for x
[tex]\begin{gathered} -\frac{1}{4}x+x-\frac{1}{2}+5=-x+x \\ \\ \frac{3}{4}x+\frac{9}{2}=0 \\ \\ \frac{3}{4}x+\frac{9}{2}-\frac{9}{2}=-\frac{9}{2} \\ \\ \frac{3}{4}x=-\frac{9}{2} \\ \\ x=\frac{4}{3}\cdot(-\frac{9}{2}) \end{gathered}[/tex]So we have that
[tex]x=-\frac{36}{6}=-6[/tex]So the answer is x = - 6.
For the function f(x) = -1/2x -7,a) Evaluate the function for the following Domain:{-2,0,6}.b) Is this function linear or quadratic? Justify your answer.c) Will the graph of this function appear as a line or a parabola?
a) Evaluating the given function on the domain {-2,0,6} we get:
[tex]\begin{gathered} f(-2)=-\frac{1}{2}(-2)-7=1-7=-6 \\ f(0)=-\frac{1}{2}(0)-7=0-7=-7 \\ f(6)=-\frac{1}{2}(6)-7=-3-7=-10 \end{gathered}[/tex]b) Notice that the given function has the form:
[tex]y=mx+b[/tex]Therefore, f(x) is a linear function.
c) From the previous point we know that f(x) is a linear function therefore its graph is a line.
The Cooking Club made some pies to sell ata basketball game to raise money for thenew math books. The cafeteria contributedfour pies to the sale. Each pie was then cutinto five pieces and sold. There were a totalof 60 pieces to sell. How many pies did theclub make?
SOLUTION
We know that the cooking club made pies to sell, but we don't know how many pies the cooking club made. So, let the number of pies the cooking club made be x.
Now the cafeteria contributed 4 pies that were cut into 5 pieces and sold. So the cafeteria contributed
[tex]4\text{ pies }\times5\text{ pieces = 20 pieces }[/tex]The cafeteria contributed 20 pieces.
Also we were told that a total of 60 pieces were sold. So what the club sold should be 60 pieces in total minus what the cafeteria sold.
So we have
[tex]60-20=40\text{ pieces }[/tex]The club sold 40 pieces.
Now remember that a pie is cut into 5 pieces, so this means that 40 pieces should make
[tex]\frac{40}{5}=8\text{ pies}[/tex]Hence the club made 8 pies
Hi, can you help me answer this question please, thank you
Using the 68-95-99.7% rule;
[tex]\begin{gathered} z=\frac{x-\mu}{\sigma} \\ z=\frac{11-35}{8} \\ z=-\frac{24}{3} \\ z=-3 \\ z=\frac{27-35}{8} \\ z=-\frac{8}{8} \\ z=-1 \\ \end{gathered}[/tex]Then, we have;
Thus;
[tex]P(-3Hence, the approximate percentage is;[tex]15.73[/tex]Graph the parabola Y = X^2 + 6x +13 Play five points on a parabola the vertex two points to the left of the vertex and two points to the right of the vertex then click on the graph a function button
Given:
[tex]y=x^2+6x+13[/tex]To plot 5 points on the parabola:
The given equation can be written as,
[tex]\begin{gathered} y=x^2+6x++3^2-3^2+13 \\ y=(x+3)^2-9+13 \\ y=(x+3)^2+4 \end{gathered}[/tex]Therefore, the vertex of the parabola is (-3, 4).
Put x=-2, we get y=5. So, the point is (-2, 5).
Put x=-1, we get y=8. So, the point is (-1, 8).
Put x=-4, we get y=5. So, the point is (-4, 5).
Put x=-5, we get y=8. So, the point is (-5, 8).
So, the graph is,
Check number 1 please and help me solve the rest.
In this session we will focus on question 5. We are asked to simplify the following expression
[tex](\frac{\placeholder{⬚}\text{ -1}}{2}x^3y^0)(6x^2y^3)(\text{ -3y\rparen}[/tex]First, recall that any non zero number, raised to the 0th power is equal to 1. So we have
[tex]y^0=1[/tex]Also, recall the property of exponents that given a,b,c and we have that
[tex]a^b\cdot a^c=a^{b+c}[/tex]What we will do is that we are going to group the expression into another expressions, organized by the same variable and the constants. So we get
[tex](\frac{\placeholder{⬚}\text{ -1}}{2}\cdot6\cdot(\text{ -}3))(x^3\cdot x^2)(y^3\cdot y)[/tex]so applying the property on each expression, and operating, we get
[tex](9)(x^5)(y^4)=9x^5y^4[/tex]How would you write this in written form as an equation?
Explanation
We are given the parent function:
[tex]f(x)=\log_x[/tex]First, the horizontal shift 6 spaces to the left follows the rule:
[tex]f(x)\to f(x+6)[/tex]Next, the vertical shift 2 spaces down follows the rule:
[tex]\begin{gathered} f(x)\to f(x)-2 \\ f(x+6)\to f(x+6)-2 \end{gathered}[/tex]Finally, the reflection over y = k can be represented as:
[tex]\begin{gathered} Let\text{ }g(x)\text{ }be\text{ }the\text{ }combined\text{ }function \\ g(x)=2k+2-f(x+6) \\ \therefore g(x)=2k+2-log(x+6) \end{gathered}[/tex]Suppose k = 2, the graph becomes:
The red curve is the reflected curve.
Hence, the answer is:
[tex]\begin{equation*} g(x)=2k+2-log(x+6) \end{equation*}[/tex]What are the coordinates of the midpoint of AB?이- 1]ㅇ(-1,-)ㅇ(--)
The midpoint formula
[tex](x_{\frac{1}{2}},y_{\frac{1}{2}})=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2_{}})[/tex]then, apply the formula
[tex]\begin{gathered} (x_{\frac{1}{2}},y_{\frac{1}{2}})=(\frac{-4+1}{2},\frac{-4+2}{2}) \\ (x_{\frac{1}{2}},y_{\frac{1}{2}})=(\frac{-3}{2},\frac{-2}{2}) \\ (x_{\frac{1}{2}},y_{\frac{1}{2}})=(\frac{-3}{2},-1) \end{gathered}[/tex]i am stuck on this question, any help would be greatly appreciated The indicated line is the 3/5x+17 and is passing through the points -5, 15
We want to find a parallel line to y = (3/5)x + 17 passing through the point (-5,15)
Any two lines are parallel if they have the same slope.
Therefore, the line we are looking for must have slope 3/5
If the line passes through (-5,15), its slope s must be given by:
[tex]\begin{gathered} 15=\frac{3}{5}x+s \\ 15=\frac{3}{5}(-5)+s \\ 15=-3+s \\ s=18 \end{gathered}[/tex]Therefore, the line is given by:
[tex]y=\frac{3}{5}x+18[/tex]the measure of angles 153 what is the measure of its supplement angle
For 2 angles to be supplementary, the sum of its measures must be 180.
The supplement angle to 153 is found by substracting 153 to 180.
[tex]\begin{gathered} \measuredangle s+153=180 \\ \measuredangle s=180-153 \\ \measuredangle s=27 \end{gathered}[/tex]The supplement angle to 153 is 27.
how do I Write the explicit function for a geometric sequence with recursive functionare gn = gn-1 × 4; g1 = 2.
g1 =