Answer:
0.5
Step-by-step explanation:
The given ordered pairs are: (1,2) (2,2.5) (3,3) (4,3.5) (5,4)
The corresponding sequence is :
2,2.5,3,3.5, 4,....
We can see that there is a common difference of d=0.5.
Better still we could use the slope formula:
![m = \frac{y_2-y_1}{x_2-x_1}](https://tex.z-dn.net/?f=m%20%3D%20%20%5Cfrac%7By_2-y_1%7D%7Bx_2-x_1%7D%20)
We substitute the point (1,2) and (2,2.5).
The rate of change of this sequence is:
![m = \frac{2.5 - 2}{2 - 1} = \frac{0.5}{1} = 0.5](https://tex.z-dn.net/?f=m%20%3D%20%20%5Cfrac%7B2.5%20-%202%7D%7B2%20-%201%7D%20%20%3D%20%20%5Cfrac%7B0.5%7D%7B1%7D%20%20%3D%200.5)
Answer:
<em>The solution is too long. So, I included them in the explanation</em>
Step-by-step explanation:
This question has missing details. However, I've corrected each question before solving them
Required: Determine the inverse
1:
![f(x) = 25x - 18](https://tex.z-dn.net/?f=f%28x%29%20%3D%2025x%20-%2018)
Replace f(x) with y
![y = 25x - 18](https://tex.z-dn.net/?f=y%20%3D%2025x%20-%2018)
Swap y & x
![x = 25y - 18](https://tex.z-dn.net/?f=x%20%3D%2025y%20-%2018)
![x + 18 = 25y - 18 + 18](https://tex.z-dn.net/?f=x%20%2B%2018%20%3D%2025y%20-%2018%20%2B%2018)
![x + 18 = 25y](https://tex.z-dn.net/?f=x%20%2B%2018%20%3D%2025y)
Divide through by 25
![\frac{x + 18}{25} = y](https://tex.z-dn.net/?f=%5Cfrac%7Bx%20%2B%2018%7D%7B25%7D%20%3D%20y)
![y = \frac{x + 18}{25}](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7Bx%20%2B%2018%7D%7B25%7D)
Replace y with f'(x)
![f'(x) = \frac{x + 18}{25}](https://tex.z-dn.net/?f=f%27%28x%29%20%3D%20%5Cfrac%7Bx%20%2B%2018%7D%7B25%7D)
2. ![g(x) = \frac{12x - 1}{7}](https://tex.z-dn.net/?f=g%28x%29%20%3D%20%5Cfrac%7B12x%20-%201%7D%7B7%7D)
Replace g(x) with y
![y = \frac{12x - 1}{7}](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B12x%20-%201%7D%7B7%7D)
Swap y & x
![x = \frac{12y - 1}{7}](https://tex.z-dn.net/?f=x%20%3D%20%5Cfrac%7B12y%20-%201%7D%7B7%7D)
![7x = 12y - 1](https://tex.z-dn.net/?f=7x%20%3D%2012y%20-%201)
Add 1 to both sides
![7x +1 = 12y - 1 + 1](https://tex.z-dn.net/?f=7x%20%2B1%20%3D%2012y%20-%201%20%2B%201)
![7x +1 = 12y](https://tex.z-dn.net/?f=7x%20%2B1%20%3D%2012y)
Make y the subject
![y = \frac{7x + 1}{12}](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B7x%20%2B%201%7D%7B12%7D)
![g'(x) = \frac{7x + 1}{12}](https://tex.z-dn.net/?f=g%27%28x%29%20%3D%20%5Cfrac%7B7x%20%2B%201%7D%7B12%7D)
3: ![h(x) = -\frac{9x}{4} - \frac{1}{3}](https://tex.z-dn.net/?f=h%28x%29%20%3D%20-%5Cfrac%7B9x%7D%7B4%7D%20-%20%5Cfrac%7B1%7D%7B3%7D)
Replace h(x) with y
![y = -\frac{9x}{4} - \frac{1}{3}](https://tex.z-dn.net/?f=y%20%3D%20-%5Cfrac%7B9x%7D%7B4%7D%20-%20%5Cfrac%7B1%7D%7B3%7D)
Swap y & x
![x = -\frac{9y}{4} - \frac{1}{3}](https://tex.z-dn.net/?f=x%20%3D%20-%5Cfrac%7B9y%7D%7B4%7D%20-%20%5Cfrac%7B1%7D%7B3%7D)
Add
to both sides
![x + \frac{1}{3}= -\frac{9y}{4} - \frac{1}{3} + \frac{1}{3}](https://tex.z-dn.net/?f=x%20%2B%20%5Cfrac%7B1%7D%7B3%7D%3D%20-%5Cfrac%7B9y%7D%7B4%7D%20-%20%5Cfrac%7B1%7D%7B3%7D%20%2B%20%5Cfrac%7B1%7D%7B3%7D)
![x + \frac{1}{3}= -\frac{9y}{4}](https://tex.z-dn.net/?f=x%20%2B%20%5Cfrac%7B1%7D%7B3%7D%3D%20-%5Cfrac%7B9y%7D%7B4%7D)
Multiply through by -4
![-4(x + \frac{1}{3})= -4(-\frac{9y}{4})](https://tex.z-dn.net/?f=-4%28x%20%2B%20%5Cfrac%7B1%7D%7B3%7D%29%3D%20-4%28-%5Cfrac%7B9y%7D%7B4%7D%29)
![-4x - \frac{4}{3}= 9y](https://tex.z-dn.net/?f=-4x%20-%20%5Cfrac%7B4%7D%7B3%7D%3D%209y)
Divide through by 9
![(-4x - \frac{4}{3})/9= y](https://tex.z-dn.net/?f=%28-4x%20-%20%5Cfrac%7B4%7D%7B3%7D%29%2F9%3D%20y)
![-4x * \frac{1}{9} - \frac{4}{3} * \frac{1}{9} = y](https://tex.z-dn.net/?f=-4x%20%2A%20%5Cfrac%7B1%7D%7B9%7D%20-%20%5Cfrac%7B4%7D%7B3%7D%20%2A%20%5Cfrac%7B1%7D%7B9%7D%20%3D%20y)
![\frac{-4x}{9} - \frac{4}{27}= y](https://tex.z-dn.net/?f=%5Cfrac%7B-4x%7D%7B9%7D%20-%20%5Cfrac%7B4%7D%7B27%7D%3D%20y)
![y = \frac{-4x}{9} - \frac{4}{27}](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B-4x%7D%7B9%7D%20-%20%5Cfrac%7B4%7D%7B27%7D)
![h'(x) = \frac{-4x}{9} - \frac{4}{27}](https://tex.z-dn.net/?f=h%27%28x%29%20%3D%20%5Cfrac%7B-4x%7D%7B9%7D%20-%20%5Cfrac%7B4%7D%7B27%7D)
4:
![f(x) = x^9](https://tex.z-dn.net/?f=f%28x%29%20%3D%20x%5E9)
Replace f(x) with y
![y = x^9](https://tex.z-dn.net/?f=y%20%3D%20x%5E9)
Swap y with x
![x = y^9](https://tex.z-dn.net/?f=x%20%3D%20y%5E9)
Take 9th root
![x^{\frac{1}{9}} = y](https://tex.z-dn.net/?f=x%5E%7B%5Cfrac%7B1%7D%7B9%7D%7D%20%3D%20y)
![y = x^{\frac{1}{9}}](https://tex.z-dn.net/?f=y%20%3D%20x%5E%7B%5Cfrac%7B1%7D%7B9%7D%7D)
Replace y with f'(x)
![f'(x) = x^{\frac{1}{9}}](https://tex.z-dn.net/?f=f%27%28x%29%20%3D%20x%5E%7B%5Cfrac%7B1%7D%7B9%7D%7D)
5:
![f(a) = a^3 + 8](https://tex.z-dn.net/?f=f%28a%29%20%3D%20a%5E3%20%2B%208)
Replace f(a) with y
![y = a^3 + 8](https://tex.z-dn.net/?f=y%20%3D%20a%5E3%20%2B%208)
Swap a with y
![a = y^3 + 8](https://tex.z-dn.net/?f=a%20%3D%20y%5E3%20%2B%208)
Subtract 8
![a - 8 = y^3 + 8 - 8](https://tex.z-dn.net/?f=a%20-%208%20%3D%20y%5E3%20%2B%208%20-%208)
![a - 8 = y^3](https://tex.z-dn.net/?f=a%20-%208%20%3D%20y%5E3)
Take cube root
![\sqrt[3]{a-8} = y](https://tex.z-dn.net/?f=%5Csqrt%5B3%5D%7Ba-8%7D%20%3D%20y)
![y = \sqrt[3]{a-8}](https://tex.z-dn.net/?f=y%20%3D%20%5Csqrt%5B3%5D%7Ba-8%7D)
Replace y with f'(a)
![f'(a) = \sqrt[3]{a-8}](https://tex.z-dn.net/?f=f%27%28a%29%20%3D%20%5Csqrt%5B3%5D%7Ba-8%7D)
6:
![g(a) = a^2 + 8a- 7](https://tex.z-dn.net/?f=g%28a%29%20%3D%20a%5E2%20%2B%208a-%207)
Replace g(a) with y
![y = a^2 + 8a - 7](https://tex.z-dn.net/?f=y%20%3D%20a%5E2%20%2B%208a%20-%207)
Swap positions of y and a
![a = y^2 + 8y - 7](https://tex.z-dn.net/?f=a%20%3D%20y%5E2%20%2B%208y%20-%207)
![y^2 + 8y - 7 - a = 0](https://tex.z-dn.net/?f=y%5E2%20%2B%208y%20-%207%20-%20a%20%3D%200)
Solve using quadratic formula:
![y = \frac{-b\±\sqrt{b^2 - 4ac}}{2a}](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B-b%5C%C2%B1%5Csqrt%7Bb%5E2%20-%204ac%7D%7D%7B2a%7D)
;
; ![c = -7 - a](https://tex.z-dn.net/?f=c%20%3D%20-7%20-%20a)
becomes
![y = \frac{-8 \±\sqrt{8^2 - 4 * 1 * (-7-a)}}{2 * 1}](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B-8%20%5C%C2%B1%5Csqrt%7B8%5E2%20-%204%20%2A%201%20%2A%20%28-7-a%29%7D%7D%7B2%20%2A%201%7D)
![y = \frac{-8 \±\sqrt{64 + 28 + 4a}}{2 * 1}](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B-8%20%5C%C2%B1%5Csqrt%7B64%20%2B%2028%20%2B%204a%7D%7D%7B2%20%2A%201%7D)
![y = \frac{-8 \±\sqrt{92 + 4a}}{2 * 1}](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B-8%20%5C%C2%B1%5Csqrt%7B92%20%2B%204a%7D%7D%7B2%20%2A%201%7D)
![y = \frac{-8 \±\sqrt{92 + 4a}}{2 }](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B-8%20%5C%C2%B1%5Csqrt%7B92%20%2B%204a%7D%7D%7B2%20%7D)
Factorize
![y = \frac{-8 \±\sqrt{4(23 + a)}}{2 }](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B-8%20%5C%C2%B1%5Csqrt%7B4%2823%20%2B%20a%29%7D%7D%7B2%20%7D)
![y = \frac{-8 \±2\sqrt{(23 + a)}}{2 }](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B-8%20%5C%C2%B12%5Csqrt%7B%2823%20%2B%20a%29%7D%7D%7B2%20%7D)
![y = -4 \±\sqrt{(23 + a)}](https://tex.z-dn.net/?f=y%20%3D%20-4%20%5C%C2%B1%5Csqrt%7B%2823%20%2B%20a%29%7D)
![g'(a) = -4 \±\sqrt{(23 + a)}](https://tex.z-dn.net/?f=g%27%28a%29%20%3D%20-4%20%5C%C2%B1%5Csqrt%7B%2823%20%2B%20a%29%7D)
7:
![f(b) = (b + 6)(b - 2)](https://tex.z-dn.net/?f=f%28b%29%20%3D%20%28b%20%2B%206%29%28b%20-%202%29)
Replace f(b) with y
![y = (b + 6)(b - 2)](https://tex.z-dn.net/?f=y%20%20%3D%20%28b%20%2B%206%29%28b%20-%202%29)
Swap y and b
![b = (y + 6)(y - 2)](https://tex.z-dn.net/?f=b%20%20%3D%20%28y%20%2B%206%29%28y%20-%202%29)
Open Brackets
![b = y^2 + 6y - 2y - 12](https://tex.z-dn.net/?f=b%20%20%3D%20y%5E2%20%2B%206y%20-%202y%20-%2012)
![b = y^2 + 4y - 12](https://tex.z-dn.net/?f=b%20%20%3D%20y%5E2%20%2B%204y%20-%2012)
![y^2 + 4y - 12 - b = 0](https://tex.z-dn.net/?f=y%5E2%20%2B%204y%20-%2012%20-%20b%20%3D%200)
Solve using quadratic formula:
![y = \frac{-b\±\sqrt{b^2 - 4ac}}{2a}](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B-b%5C%C2%B1%5Csqrt%7Bb%5E2%20-%204ac%7D%7D%7B2a%7D)
;
; ![c = -12 - b](https://tex.z-dn.net/?f=c%20%3D%20-12%20-%20b)
becomes
![y = \frac{-4\±\sqrt{4^2 - 4 * 1 * (-12-b)}}{2*1}](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B-4%5C%C2%B1%5Csqrt%7B4%5E2%20-%204%20%2A%201%20%2A%20%28-12-b%29%7D%7D%7B2%2A1%7D)
![y = \frac{-4\±\sqrt{4^2 - 4 *(-12-b)}}{2}](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B-4%5C%C2%B1%5Csqrt%7B4%5E2%20-%204%20%2A%28-12-b%29%7D%7D%7B2%7D)
Factorize:
![y = \frac{-4\±\sqrt{4(4 - (-12-b))}}{2}](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B-4%5C%C2%B1%5Csqrt%7B4%284%20-%20%28-12-b%29%29%7D%7D%7B2%7D)
![y = \frac{-4\±2\sqrt{(4 - (-12-b))}}{2}](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B-4%5C%C2%B12%5Csqrt%7B%284%20-%20%28-12-b%29%29%7D%7D%7B2%7D)
![y = \frac{-4\±2\sqrt{(4 +12+b)}}{2}](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B-4%5C%C2%B12%5Csqrt%7B%284%20%2B12%2Bb%29%7D%7D%7B2%7D)
![y = \frac{-4\±2\sqrt{16+b}}{2}](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B-4%5C%C2%B12%5Csqrt%7B16%2Bb%7D%7D%7B2%7D)
![y = -2\±\sqrt{16+b}](https://tex.z-dn.net/?f=y%20%3D%20-2%5C%C2%B1%5Csqrt%7B16%2Bb%7D)
Replace y with f'(b)
![f'(b) = -2\±\sqrt{16+b}](https://tex.z-dn.net/?f=f%27%28b%29%20%3D%20-2%5C%C2%B1%5Csqrt%7B16%2Bb%7D)
8:
![h(x) = \frac{2x+17}{3x+1}](https://tex.z-dn.net/?f=h%28x%29%20%3D%20%5Cfrac%7B2x%2B17%7D%7B3x%2B1%7D)
Replace h(x) with y
![y = \frac{2x+17}{3x+1}](https://tex.z-dn.net/?f=y%20%20%3D%20%5Cfrac%7B2x%2B17%7D%7B3x%2B1%7D)
Swap x and y
![x = \frac{2y+17}{3y+1}](https://tex.z-dn.net/?f=x%20%20%3D%20%5Cfrac%7B2y%2B17%7D%7B3y%2B1%7D)
Cross Multiply
![(3y + 1)x = 2y + 17](https://tex.z-dn.net/?f=%283y%20%2B%201%29x%20%3D%202y%20%2B%2017)
![3yx + x = 2y + 17](https://tex.z-dn.net/?f=3yx%20%2B%20x%20%3D%202y%20%2B%2017)
Subtract x from both sides:
![3yx + x -x= 2y + 17-x](https://tex.z-dn.net/?f=3yx%20%2B%20x%20-x%3D%202y%20%2B%2017-x)
![3yx = 2y + 17-x](https://tex.z-dn.net/?f=3yx%20%3D%202y%20%2B%2017-x)
Subtract 2y from both sides
![3yx-2y =17-x](https://tex.z-dn.net/?f=3yx-2y%20%20%3D17-x)
Factorize:
![y(3x-2) =17-x](https://tex.z-dn.net/?f=y%283x-2%29%20%20%3D17-x)
Make y the subject
![y = \frac{17 - x}{3x - 2}](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B17%20-%20x%7D%7B3x%20-%202%7D)
Replace y with h'(x)
![h'(x) = \frac{17 - x}{3x - 2}](https://tex.z-dn.net/?f=h%27%28x%29%20%3D%20%5Cfrac%7B17%20-%20x%7D%7B3x%20-%202%7D)
9:
![h(c) = \sqrt{2c + 2}](https://tex.z-dn.net/?f=h%28c%29%20%3D%20%5Csqrt%7B2c%20%2B%202%7D)
Replace h(c) with y
![y = \sqrt{2c + 2}](https://tex.z-dn.net/?f=y%20%3D%20%5Csqrt%7B2c%20%2B%202%7D)
Swap positions of y and c
![c = \sqrt{2y + 2}](https://tex.z-dn.net/?f=c%20%3D%20%5Csqrt%7B2y%20%2B%202%7D)
Square both sides
![c^2 = 2y + 2](https://tex.z-dn.net/?f=c%5E2%20%3D%202y%20%2B%202)
Subtract 2 from both sides
![c^2 - 2= 2y](https://tex.z-dn.net/?f=c%5E2%20-%202%3D%202y)
Make y the subject
![y = \frac{c^2 - 2}{2}](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7Bc%5E2%20-%202%7D%7B2%7D)
![h'(c) = \frac{c^2 - 2}{2}](https://tex.z-dn.net/?f=h%27%28c%29%20%3D%20%5Cfrac%7Bc%5E2%20-%202%7D%7B2%7D)
10:
![f(x) = \frac{x + 10}{9x - 1}](https://tex.z-dn.net/?f=f%28x%29%20%3D%20%5Cfrac%7Bx%20%2B%2010%7D%7B9x%20-%201%7D)
Replace f(x) with y
![y = \frac{x + 10}{9x - 1}](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7Bx%20%2B%2010%7D%7B9x%20-%201%7D)
Swap positions of x and y
![x = \frac{y + 10}{9y - 1}](https://tex.z-dn.net/?f=x%20%3D%20%5Cfrac%7By%20%2B%2010%7D%7B9y%20-%201%7D)
Cross Multiply
![x(9y - 1) = y + 10](https://tex.z-dn.net/?f=x%289y%20-%201%29%20%3D%20y%20%2B%2010)
![9xy - x = y + 10](https://tex.z-dn.net/?f=9xy%20-%20x%20%3D%20y%20%2B%2010)
Subtract y from both sides
![9xy - y - x = y - y+ 10](https://tex.z-dn.net/?f=9xy%20-%20y%20-%20x%20%3D%20y%20-%20y%2B%2010)
![9xy - y - x = 10](https://tex.z-dn.net/?f=9xy%20-%20y%20-%20x%20%3D%20%2010)
Add x to both sides
![9xy - y - x + x= 10 + x](https://tex.z-dn.net/?f=9xy%20-%20y%20-%20x%20%2B%20x%3D%20%2010%20%2B%20x)
![9xy - y = 10 + x](https://tex.z-dn.net/?f=9xy%20-%20y%20%3D%20%2010%20%2B%20x)
Factorize
![y(9x - 1) = 10 + x](https://tex.z-dn.net/?f=y%289x%20-%201%29%20%3D%20%2010%20%2B%20x)
Make y the subject
![y = \frac{10 + x}{9x - 1}](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B10%20%2B%20x%7D%7B9x%20-%201%7D)
Replace y with f'(x)
![f'(x) = \frac{10 + x}{9x -1}](https://tex.z-dn.net/?f=f%27%28x%29%20%3D%20%5Cfrac%7B10%20%2B%20x%7D%7B9x%20-1%7D)
Answer:
-1
Step-by-step explanation:
-7 + 6 = -1
Answer: 0.6 = 0.60 and 39 is lower than 60 so there for 0.6 is higher
Step-by-step explanation:
To find the surface area of the side of the soup can find the circumference first, the circumference is 2