Answer: a) ![\frac{5}{2}x+10=f^{-1}(x)=g(x)](https://tex.z-dn.net/?f=%5Cfrac%7B5%7D%7B2%7Dx%2B10%3Df%5E%7B-1%7D%28x%29%3Dg%28x%29)
Step-by-step explanation:
Since we have given that
![f(x)=\frac{2}{5}x-4](https://tex.z-dn.net/?f=f%28x%29%3D%5Cfrac%7B2%7D%7B5%7Dx-4)
a.) Find the inverse of f(x) and name it g(x).
Let f(x) = y
So, it becomes
Switching x to y , we get
![x=\frac{2}{5}y-4](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B2%7D%7B5%7Dy-4)
![5x=2y-20\\\\5x+20=2y\\\\\frac{5x+20}{2}=y\\\\\frac{5}{2}x+10=y\\\\\frac{5}{2}x+10=f^{-1}(x)=g(x)](https://tex.z-dn.net/?f=5x%3D2y-20%5C%5C%5C%5C5x%2B20%3D2y%5C%5C%5C%5C%5Cfrac%7B5x%2B20%7D%7B2%7D%3Dy%5C%5C%5C%5C%5Cfrac%7B5%7D%7B2%7Dx%2B10%3Dy%5C%5C%5C%5C%5Cfrac%7B5%7D%7B2%7Dx%2B10%3Df%5E%7B-1%7D%28x%29%3Dg%28x%29)
b) . Use composition to show that f(x) and g(x) are inverses of each other.
![\mathrm{For}\:f=\frac{2}{5}x-4\:\\\\\mathrm{substitute}\:x\:\mathrm{with}\:g\left(x\right)=\frac{5}{2}x+10\\\\=\frac{2}{5}\left(\frac{5}{2}x+10\right)-4\\\\=x](https://tex.z-dn.net/?f=%5Cmathrm%7BFor%7D%5C%3Af%3D%5Cfrac%7B2%7D%7B5%7Dx-4%5C%3A%5C%5C%5C%5C%5Cmathrm%7Bsubstitute%7D%5C%3Ax%5C%3A%5Cmathrm%7Bwith%7D%5C%3Ag%5Cleft%28x%5Cright%29%3D%5Cfrac%7B5%7D%7B2%7Dx%2B10%5C%5C%5C%5C%3D%5Cfrac%7B2%7D%7B5%7D%5Cleft%28%5Cfrac%7B5%7D%7B2%7Dx%2B10%5Cright%29-4%5C%5C%5C%5C%3Dx)
Similarly,
![\mathrm{g\left(x\right)=\frac{5}{2}x+10,\:f\left(x\right)=\frac{2}{5}x-4,\:g\left(x\right)\circ \:f\left(x\right)}\\\\\mathrm{For}\:g=\frac{5}{2}x+10\:\mathrm{substitute}\:x\:\mathrm{with}\:f\left(x\right)=\frac{2}{5}x-4\\\\=\frac{5}{2}\left(\frac{2}{5}x-4\right)+10\\\\=x](https://tex.z-dn.net/?f=%5Cmathrm%7Bg%5Cleft%28x%5Cright%29%3D%5Cfrac%7B5%7D%7B2%7Dx%2B10%2C%5C%3Af%5Cleft%28x%5Cright%29%3D%5Cfrac%7B2%7D%7B5%7Dx-4%2C%5C%3Ag%5Cleft%28x%5Cright%29%5Ccirc%20%5C%3Af%5Cleft%28x%5Cright%29%7D%5C%5C%5C%5C%5Cmathrm%7BFor%7D%5C%3Ag%3D%5Cfrac%7B5%7D%7B2%7Dx%2B10%5C%3A%5Cmathrm%7Bsubstitute%7D%5C%3Ax%5C%3A%5Cmathrm%7Bwith%7D%5C%3Af%5Cleft%28x%5Cright%29%3D%5Cfrac%7B2%7D%7B5%7Dx-4%5C%5C%5C%5C%3D%5Cfrac%7B5%7D%7B2%7D%5Cleft%28%5Cfrac%7B2%7D%7B5%7Dx-4%5Cright%29%2B10%5C%5C%5C%5C%3Dx)
so, both are inverses of each other.
c) Draw the graphs of f(x) and g(x) on the same coordinate plane.
As shown below in the graph , Since for inverse function we need an axis of symmetry i.e. y=x
And both f(x) and g(x) are symmetry to y=x.
∴ f(x) and g(x) are inverses of each other.
Answer:
c)55
Step-by-step explanation:
Subtract the volume of the water from the volume of the tank to get the total volume of the marbles.
26 liters - 20.9 liters = 5.1 liters
He has 20.9 liters of water.
He has 85 marbles that add up to 5.1 liters of volume.
To find the volume of one marble, we divide 5.1 liters by 85.
(5.1 liters)/85 = 0.06 liters
The volume of each marble is 0.06 liters.
If he uses 200 marbles, the the marbles will occupy 200 * 0.06 liters = 12 liters.
The volume of the tank is 26 liters. 12 liters of volume is occupied by the marbles. The rest of the volume is water.
26 liters - 12 liters = 14 liters
<h3>
Answer: 15 cubic feet</h3>
Explanation:
18 inches = 18/12 = 1.5 feet
The cabinet is 5 ft by 1.5 ft by 2 ft. Multiply the values to get 5*1.5*2 = 15