Answer:
Domain and Range of g(f(x)) are 'All real numbers' and {y | y>6 } respectively
Step-by-step explanation:
We have the functions, f(x) = eˣ and g(x) = x+6
So, their composition will be g(f(x)).
Then, g(f(x)) = g(eˣ) = eˣ+6
Thus, g(f(x)) = eˣ+6.
Since the domain and range of f(x) = eˣ are all real numbers and positive real numbers respectively.
Moreover, the function g(f(x)) = eˣ+6 is the function f(x) translated up by 6 units.
Hence, the domain and range of g(f(x)) are 'All real numbers' and {y | y>6 } respectively.
Answer:
Option A)
![h=\frac{(3v-2)}{(v-1)}](https://tex.z-dn.net/?f=h%3D%5Cfrac%7B%283v-2%29%7D%7B%28v-1%29%7D)
Step-by-step explanation:
Remember that the volume of a rectangle is:
![V = lwh](https://tex.z-dn.net/?f=V%20%3D%20lwh)
Where l is the length, w is the width and h is the height.
In the figure, these three dimensions are given as a function of the variable v.
Then the volume will be the product of the three expressions.
If we have the volume, the width and the length of the rectangle, then we find the height when we divide the volume by the product of the width and the length
![h = \frac{V}{(lw)}](https://tex.z-dn.net/?f=h%20%3D%20%5Cfrac%7BV%7D%7B%28lw%29%7D)
The volume is:
![V=\frac{3v^2-19v-14}{3v^2-v-2}](https://tex.z-dn.net/?f=V%3D%5Cfrac%7B3v%5E2-19v-14%7D%7B3v%5E2-v-2%7D)
The product of the width and the length is:
![lw=\frac{v-7}{3v+2}*\frac{3v+2}{3v-2}\\\\lw=\frac{v-7}{3v-2}](https://tex.z-dn.net/?f=lw%3D%5Cfrac%7Bv-7%7D%7B3v%2B2%7D%2A%5Cfrac%7B3v%2B2%7D%7B3v-2%7D%5C%5C%5C%5Clw%3D%5Cfrac%7Bv-7%7D%7B3v-2%7D)
Now
![h=\frac{V}{lw}=\frac{\frac{3v^2-19v-14}{3v^2-v-2}}{\frac{v-7}{3v-2}}\\\\\\h=\frac{3v^2-19v-14(3v-2)}{3v^2-v-2(v-7)}](https://tex.z-dn.net/?f=h%3D%5Cfrac%7BV%7D%7Blw%7D%3D%5Cfrac%7B%5Cfrac%7B3v%5E2-19v-14%7D%7B3v%5E2-v-2%7D%7D%7B%5Cfrac%7Bv-7%7D%7B3v-2%7D%7D%5C%5C%5C%5C%5C%5Ch%3D%5Cfrac%7B3v%5E2-19v-14%283v-2%29%7D%7B3v%5E2-v-2%28v-7%29%7D)
we factor quadratic expressions
![3v^2-19v-14\\a=3\\b=-19\\c=-14\\](https://tex.z-dn.net/?f=3v%5E2-19v-14%5C%5Ca%3D3%5C%5Cb%3D-19%5C%5Cc%3D-14%5C%5C)
We use the quadratic formula to factor the expression
![3v^2-19v-14\\\\v_1 =\frac{19+\sqrt{19^2-4(3)(-14)}}{2(3)}\\\\v_2=\frac{19-\sqrt{19^2-4(3)(-14)}}{2(3)}\\\\v_1=7\\\\v_2=-\frac{2}{3}\\\\3v^2-19v-14 = (v-7)(3v+2)](https://tex.z-dn.net/?f=3v%5E2-19v-14%5C%5C%5C%5Cv_1%20%3D%5Cfrac%7B19%2B%5Csqrt%7B19%5E2-4%283%29%28-14%29%7D%7D%7B2%283%29%7D%5C%5C%5C%5Cv_2%3D%5Cfrac%7B19-%5Csqrt%7B19%5E2-4%283%29%28-14%29%7D%7D%7B2%283%29%7D%5C%5C%5C%5Cv_1%3D7%5C%5C%5C%5Cv_2%3D-%5Cfrac%7B2%7D%7B3%7D%5C%5C%5C%5C3v%5E2-19v-14%20%3D%20%28v-7%29%283v%2B2%29)
We also factor the quadratic function
using the quadratic formula
![3v^2-v-2\\\\a=3\\\\b=-1\\\\c=-2\\\\v_1 =\frac{1+\sqrt{1^2 -4(3)(-2)}}{2(3)}\\\\v_2=\frac{1-\sqrt{1^2 -4(3)(-2)}}{2(3)}\\\\v_1=1\\\\v_2=-\frac{2}{3}\\\\3v^2-v-2 = (v-1)(3v+2)](https://tex.z-dn.net/?f=3v%5E2-v-2%5C%5C%5C%5Ca%3D3%5C%5C%5C%5Cb%3D-1%5C%5C%5C%5Cc%3D-2%5C%5C%5C%5Cv_1%20%3D%5Cfrac%7B1%2B%5Csqrt%7B1%5E2%20-4%283%29%28-2%29%7D%7D%7B2%283%29%7D%5C%5C%5C%5Cv_2%3D%5Cfrac%7B1-%5Csqrt%7B1%5E2%20-4%283%29%28-2%29%7D%7D%7B2%283%29%7D%5C%5C%5C%5Cv_1%3D1%5C%5C%5C%5Cv_2%3D-%5Cfrac%7B2%7D%7B3%7D%5C%5C%5C%5C3v%5E2-v-2%20%3D%20%28v-1%29%283v%2B2%29)
So the height is
![h=\frac{(v-7)(3v+2)(3v-2)}{(v-1)(3v+2)(v-7)}\\\\h=\frac{(3v-2)}{(v-1)}](https://tex.z-dn.net/?f=h%3D%5Cfrac%7B%28v-7%29%283v%2B2%29%283v-2%29%7D%7B%28v-1%29%283v%2B2%29%28v-7%29%7D%5C%5C%5C%5Ch%3D%5Cfrac%7B%283v-2%29%7D%7B%28v-1%29%7D)
you can measure the first angle with a protractor and then measure the other one see if there is a different angle
Step-by-step explanation:
To find the slope, calculate rise and run. The formula is as follows...
![\frac{Rise}{Run} =\frac{y2-y1}{x2-x1}](https://tex.z-dn.net/?f=%5Cfrac%7BRise%7D%7BRun%7D%20%3D%5Cfrac%7By2-y1%7D%7Bx2-x1%7D)