![\bf ~~~~~~~~~~~~\textit{function transformations} \\\\\\ % templates f(x)={{ A}}({{ B}}x+{{ C}})+{{ D}} \\\\ ~~~~y={{ A}}({{ B}}x+{{ C}})+{{ D}} \\\\ f(x)={{ A}}\sqrt{{{ B}}x+{{ C}}}+{{ D}} \\\\ f(x)={{ A}}(\mathbb{R})^{{{ B}}x+{{ C}}}+{{ D}} \\\\ f(x)={{ A}} sin\left({{ B }}x+{{ C}} \right)+{{ D}} \\\\ --------------------](https://tex.z-dn.net/?f=%5Cbf%20~~~~~~~~~~~~%5Ctextit%7Bfunction%20transformations%7D%0A%5C%5C%5C%5C%5C%5C%0A%25%20templates%0Af%28x%29%3D%7B%7B%20%20A%7D%7D%28%7B%7B%20%20B%7D%7Dx%2B%7B%7B%20%20C%7D%7D%29%2B%7B%7B%20%20D%7D%7D%0A%5C%5C%5C%5C%0A~~~~y%3D%7B%7B%20%20A%7D%7D%28%7B%7B%20%20B%7D%7Dx%2B%7B%7B%20%20C%7D%7D%29%2B%7B%7B%20%20D%7D%7D%0A%5C%5C%5C%5C%0Af%28x%29%3D%7B%7B%20%20A%7D%7D%5Csqrt%7B%7B%7B%20%20B%7D%7Dx%2B%7B%7B%20%20C%7D%7D%7D%2B%7B%7B%20%20D%7D%7D%0A%5C%5C%5C%5C%0Af%28x%29%3D%7B%7B%20%20A%7D%7D%28%5Cmathbb%7BR%7D%29%5E%7B%7B%7B%20%20B%7D%7Dx%2B%7B%7B%20%20C%7D%7D%7D%2B%7B%7B%20%20D%7D%7D%0A%5C%5C%5C%5C%0Af%28x%29%3D%7B%7B%20%20A%7D%7D%20sin%5Cleft%28%7B%7B%20B%20%7D%7Dx%2B%7B%7B%20%20C%7D%7D%20%20%5Cright%29%2B%7B%7B%20%20D%7D%7D%0A%5C%5C%5C%5C%0A--------------------)
![\bf \bullet \textit{ stretches or shrinks horizontally by } {{ A}}\cdot {{ B}}\\\\ \bullet \textit{ flips it upside-down if }{{ A}}\textit{ is negative}\\ ~~~~~~\textit{reflection over the x-axis} \\\\ \bullet \textit{ flips it sideways if }{{ B}}\textit{ is negative}\\ ~~~~~~\textit{reflection over the y-axis}](https://tex.z-dn.net/?f=%5Cbf%20%5Cbullet%20%5Ctextit%7B%20stretches%20or%20shrinks%20horizontally%20by%20%20%7D%20%7B%7B%20%20A%7D%7D%5Ccdot%20%7B%7B%20%20B%7D%7D%5C%5C%5C%5C%0A%5Cbullet%20%5Ctextit%7B%20flips%20it%20upside-down%20if%20%7D%7B%7B%20%20A%7D%7D%5Ctextit%7B%20is%20negative%7D%5C%5C%0A~~~~~~%5Ctextit%7Breflection%20over%20the%20x-axis%7D%0A%5C%5C%5C%5C%0A%5Cbullet%20%5Ctextit%7B%20flips%20it%20sideways%20if%20%7D%7B%7B%20%20B%7D%7D%5Ctextit%7B%20is%20negative%7D%5C%5C%0A~~~~~~%5Ctextit%7Breflection%20over%20the%20y-axis%7D)
![\bf \bullet \textit{ horizontal shift by }\frac{{{ C}}}{{{ B}}}\\ ~~~~~~if\ \frac{{{ C}}}{{{ B}}}\textit{ is negative, to the right}\\\\ \left. \qquad \right. if\ \frac{{{ C}}}{{{ B}}}\textit{ is positive, to the left}\\\\ \bullet \textit{ vertical shift by }{{ D}}\\ ~~~~~~if\ {{ D}}\textit{ is negative, downwards}\\\\ ~~~~~~if\ {{ D}}\textit{ is positive, upwards}\\\\ \bullet \textit{ period of }\frac{2\pi }{{{ B}}}](https://tex.z-dn.net/?f=%5Cbf%20%5Cbullet%20%5Ctextit%7B%20horizontal%20shift%20by%20%7D%5Cfrac%7B%7B%7B%20%20C%7D%7D%7D%7B%7B%7B%20%20B%7D%7D%7D%5C%5C%0A~~~~~~if%5C%20%5Cfrac%7B%7B%7B%20%20C%7D%7D%7D%7B%7B%7B%20%20B%7D%7D%7D%5Ctextit%7B%20is%20negative%2C%20to%20the%20right%7D%5C%5C%5C%5C%0A%5Cleft.%20%5Cqquad%20%20%5Cright.%20%20if%5C%20%5Cfrac%7B%7B%7B%20%20C%7D%7D%7D%7B%7B%7B%20%20B%7D%7D%7D%5Ctextit%7B%20is%20positive%2C%20to%20the%20left%7D%5C%5C%5C%5C%0A%5Cbullet%20%5Ctextit%7B%20vertical%20shift%20by%20%7D%7B%7B%20%20D%7D%7D%5C%5C%0A~~~~~~if%5C%20%7B%7B%20%20D%7D%7D%5Ctextit%7B%20is%20negative%2C%20downwards%7D%5C%5C%5C%5C%0A~~~~~~if%5C%20%7B%7B%20%20D%7D%7D%5Ctextit%7B%20is%20positive%2C%20upwards%7D%5C%5C%5C%5C%0A%5Cbullet%20%5Ctextit%7B%20period%20of%20%7D%5Cfrac%7B2%5Cpi%20%7D%7B%7B%7B%20%20B%7D%7D%7D)
with that template in mind, let's check,
![\bf f(x)=-3^{\stackrel{B}{2}x\stackrel{C}{-4}}\qquad \qquad g(x)=-3^{2x}\implies g(x)=-3^{\stackrel{B}{2}x\stackrel{C}{+0}}](https://tex.z-dn.net/?f=%5Cbf%20f%28x%29%3D-3%5E%7B%5Cstackrel%7BB%7D%7B2%7Dx%5Cstackrel%7BC%7D%7B-4%7D%7D%5Cqquad%20%5Cqquad%20g%28x%29%3D-3%5E%7B2x%7D%5Cimplies%20%20g%28x%29%3D-3%5E%7B%5Cstackrel%7BB%7D%7B2%7Dx%5Cstackrel%7BC%7D%7B%2B0%7D%7D)
notice, g(x) has a horizontal shift of C/B or +0/2, or just 0, none.
while f(x) has a horizontal shift of C/B or -4/2, or -2, to the right.
so f(x) is really just g(x), but shifted horizontally over 2 units to the right.
Here's how you solve this. So, x+y=2, right? Let's isolate x. x+y-y=2-y. x=2-y. NOW, if x=2-y, in 3x+2y=5, we can REPLACE x with 2-y and use it to solve for y! 3x+2y=5. 3(2-y)+2y=5. (3*2)+(3*-y)+2y=5. 6+(-3y)+2y=5. 6+(-y)=5. 6+(-y)-6=5-6. -y=-1. -y/-1=-1/-1. y=1.
So, if y=1, we can substitute that back into either equation--but let's go with the easier one, x+y=2. x+1=2. x+1-1=2-1. x=1. 1+1=2, so that works; let's check the other equation. 3(1)+2(1)=5. 3+2=5. 5=5. That's correct!
Answer: x=1, y=1
Answer:
![\lim_{x \to 2^-} x^3=2^3=\boxed8\\ \lim_{x \to 2^+} 3x-4=3\cdot 2-4=6-4=\boxed2](https://tex.z-dn.net/?f=%5Clim_%7Bx%20%5Cto%202%5E-%7D%20x%5E3%3D2%5E3%3D%5Cboxed8%5C%5C%20%20%5Clim_%7Bx%20%5Cto%202%5E%2B%7D%203x-4%3D3%5Ccdot%202-4%3D6-4%3D%5Cboxed2)
does not exist
Step-by-step explanation:
Inserting 2 to both formulas, you get different results. In that cases, a limit does not exist
Answer:
im pretty sure it is .17
Step-by-step explanation: