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.
Let the side length of the square be x, then A = x^2
but diagonal (z) = sqrt(2x^2)
z^2 = 2x^2
x^2 = 1/2 z^2
Thus, A = 1/2 z^2
dA/dz = 1/2 (2z) = z
The rate of change is z.
When z = 4, the rate is 4.
Answer:
its a right triangle
Step-by-step explanation:
Best jump=4.25+0.12(4.25)=4.76
best jump=75% of winning jump
Winning jump=4.76/75×100=6.35m
The answer would be the three quart bowl.Since it is clearly stated at the problem that if she 'triples' the recipe.So meaning, she needs to multiply each by three.So 1 quart = 3 quarts2 quarts = 6 quarts3 quarts = 9 quarts
So the smallest now would be the three quarts bowl.