Answer:
1.
<u>Function:</u>
![f(x)=-x^2](https://tex.z-dn.net/?f=f%28x%29%3D-x%5E2)
Domain: (-∞,∞)
Range: (-∞,0]
<u>Inverse Function:</u>
![f^{-1}(x)=\sqrt{-x} ,and\\f^{-1}(x)=-\sqrt{-x}](https://tex.z-dn.net/?f=f%5E%7B-1%7D%28x%29%3D%5Csqrt%7B-x%7D%20%2Cand%5C%5Cf%5E%7B-1%7D%28x%29%3D-%5Csqrt%7B-x%7D)
Domain: (-∞,0]
Range: (-∞,∞)
2.
<u>Function:</u>
![f(x)=5x-1](https://tex.z-dn.net/?f=f%28x%29%3D5x-1)
Domain: (-∞,∞)
Range: (-∞,∞)
<u>Inverse Function:</u>
![f^{-1}(x)=\frac{1}{5}x+\frac{1}{5}](https://tex.z-dn.net/?f=f%5E%7B-1%7D%28x%29%3D%5Cfrac%7B1%7D%7B5%7Dx%2B%5Cfrac%7B1%7D%7B5%7D)
Domain: (-∞,∞)
Range: (-∞,∞)
3.
<u>Function:</u>
![f(x)=-x+3](https://tex.z-dn.net/?f=f%28x%29%3D-x%2B3)
Domain: (-∞,∞)
Range: (-∞,∞)
<u>Inverse Function:</u>
![f^{-1}(x)=-x+3](https://tex.z-dn.net/?f=f%5E%7B-1%7D%28x%29%3D-x%2B3)
Domain: (-∞,∞)
Range: (-∞,∞)
4.
<u>Function:</u>
![f(x)=x^{2}+7](https://tex.z-dn.net/?f=f%28x%29%3Dx%5E%7B2%7D%2B7)
Domain: (-∞,∞)
Range: [7,∞)
<u>Inverse Function:</u>
![f^{-1}(x)=\sqrt{x-7}, and\\f^{-1}(x)=-\sqrt{x-7}](https://tex.z-dn.net/?f=f%5E%7B-1%7D%28x%29%3D%5Csqrt%7Bx-7%7D%2C%20and%5C%5Cf%5E%7B-1%7D%28x%29%3D-%5Csqrt%7Bx-7%7D)
Domain: [7,∞)
Range: (-∞,∞)
5.
<u>Function:</u>
![f(x)=14x-4](https://tex.z-dn.net/?f=f%28x%29%3D14x-4)
Domain: (-∞,∞)
Range: (-∞,∞)
<u>Inverse Function:</u>
![f^{-1}(x)=\frac{1}{14}x+\frac{2}{7}](https://tex.z-dn.net/?f=f%5E%7B-1%7D%28x%29%3D%5Cfrac%7B1%7D%7B14%7Dx%2B%5Cfrac%7B2%7D%7B7%7D)
Domain: (-∞,∞)
Range: (-∞,∞)
6.
<u>Function:</u>
![f(x)=-3x+8](https://tex.z-dn.net/?f=f%28x%29%3D-3x%2B8)
Domain: (-∞,∞)
Range: (-∞,∞)
<u>Inverse Function:</u>
![f^{-1}(x)=-\frac{1}{3}x+\frac{8}{3}](https://tex.z-dn.net/?f=f%5E%7B-1%7D%28x%29%3D-%5Cfrac%7B1%7D%7B3%7Dx%2B%5Cfrac%7B8%7D%7B3%7D)
Domain: (-∞,∞)
Range: (-∞,∞)
Step-by-step explanation:
To find inverse of a function f(x), there are 4 steps we need to follow:
1. Replace f(x) with y
2. Interchange the y and x
3. Solve for the "new" y
4. Replace the "new" y with the notation for inverse function, ![f^{-1}(x)](https://tex.z-dn.net/?f=f%5E%7B-1%7D%28x%29)
<u>Note:</u> The domain of the original function f(x) is the range of the inverse and the range of the original function is the domain of the inverse function.
<u><em>Let's calculate each of these.</em></u>
1.
![f(x)=-x^2](https://tex.z-dn.net/?f=f%28x%29%3D-x%5E2)
Domain: There is no restriction on values of x we can put on it. Hence domain is (-∞,∞)
Range: No matter what we put into x, the y values will always be negative. And if we put 0, y value would be 0. So range is (-∞,0]
<u>Finding the inverse:</u>
![f(x)=-x^2\\y=-x^2\\x=-y^2\\y^2=-x\\y=+-\sqrt{-x} \\y=\sqrt{-x}, -\sqrt{-x}](https://tex.z-dn.net/?f=f%28x%29%3D-x%5E2%5C%5Cy%3D-x%5E2%5C%5Cx%3D-y%5E2%5C%5Cy%5E2%3D-x%5C%5Cy%3D%2B-%5Csqrt%7B-x%7D%20%5C%5Cy%3D%5Csqrt%7B-x%7D%2C%20-%5Csqrt%7B-x%7D)
So
![f^{-1}(x)=\sqrt{-x} ,and\\f^{-1}(x)=-\sqrt{-x}](https://tex.z-dn.net/?f=f%5E%7B-1%7D%28x%29%3D%5Csqrt%7B-x%7D%20%2Cand%5C%5Cf%5E%7B-1%7D%28x%29%3D-%5Csqrt%7B-x%7D)
Domain: this is the range of the original so domain is (-∞,0]
Range: this is the domain of the original so range is (-∞,∞)
2.
![f(x)=5x-1](https://tex.z-dn.net/?f=f%28x%29%3D5x-1)
Domain: There is no restriction on values of x we can put on it. Hence domain is (-∞,∞)
Range: All sorts of y values will occur, so the range is (-∞,∞)
<u>Finding the inverse:</u>
![f(x)=5x-1\\y=5x-1\\x=5y-1\\5y=x+1\\y=\frac{1}{5}x+\frac{1}{5}](https://tex.z-dn.net/?f=f%28x%29%3D5x-1%5C%5Cy%3D5x-1%5C%5Cx%3D5y-1%5C%5C5y%3Dx%2B1%5C%5Cy%3D%5Cfrac%7B1%7D%7B5%7Dx%2B%5Cfrac%7B1%7D%7B5%7D)
So
![f^{-1}(x)=\frac{1}{5}x+\frac{1}{5}](https://tex.z-dn.net/?f=f%5E%7B-1%7D%28x%29%3D%5Cfrac%7B1%7D%7B5%7Dx%2B%5Cfrac%7B1%7D%7B5%7D)
Domain: this is the range of the original so domain is (-∞,∞)
Range: this is the domain of the original so range is (-∞,∞)
3.
![f(x)=-x+3](https://tex.z-dn.net/?f=f%28x%29%3D-x%2B3)
Domain: There is no restriction on values of x we can put on it. Hence domain is (-∞,∞)
Range: All sorts of y values will occur, so the range is (-∞,∞)
<u>Finding the inverse:</u>
![f(x)=-x+3\\y=-x+3\\x=-y+3\\y=-x+3](https://tex.z-dn.net/?f=f%28x%29%3D-x%2B3%5C%5Cy%3D-x%2B3%5C%5Cx%3D-y%2B3%5C%5Cy%3D-x%2B3)
So
![f^{-1}(x)=-x+3](https://tex.z-dn.net/?f=f%5E%7B-1%7D%28x%29%3D-x%2B3)
Domain: this is the range of the original so domain is (-∞,∞)
Range: this is the domain of the original so range is (-∞,∞)
4.
![f(x)=x^{2}+7](https://tex.z-dn.net/?f=f%28x%29%3Dx%5E%7B2%7D%2B7)
Domain: There is no restriction on values of x we can put on it. Hence domain is (-∞,∞)
Range: no matter what we put into x, it will always be a positive number greater than 7. Only when we put in 0, y will be 7. So 7 is the lowest number and it can go to infinity. Hence the range is [7,∞)
<u>Finding the inverse:</u>
![f(x)=x^2+7\\y=x^2+7\\x=y^2+7\\y^2=x-7\\y=+-\sqrt{x-7}](https://tex.z-dn.net/?f=f%28x%29%3Dx%5E2%2B7%5C%5Cy%3Dx%5E2%2B7%5C%5Cx%3Dy%5E2%2B7%5C%5Cy%5E2%3Dx-7%5C%5Cy%3D%2B-%5Csqrt%7Bx-7%7D)
So
![f^{-1}(x)=\sqrt{x-7}, and\\f^{-1}(x)=-\sqrt{x-7}](https://tex.z-dn.net/?f=f%5E%7B-1%7D%28x%29%3D%5Csqrt%7Bx-7%7D%2C%20and%5C%5Cf%5E%7B-1%7D%28x%29%3D-%5Csqrt%7Bx-7%7D)
Domain: this is the range of the original so domain is [7,∞)
Range: this is the domain of the original so range is (-∞,∞)
5.
![f(x)=14x-4](https://tex.z-dn.net/?f=f%28x%29%3D14x-4)
Domain: There is no restriction on values of x we can put on it. Hence domain is (-∞,∞)
Range: no matter what we put into x, we can get any y value from negative infinity to positive infinity. So range is (-∞,∞)
<u>Finding the inverse:</u>
![f(x)=14x-4\\y=14x-4\\x=14y-4\\14y=x+4\\y=\frac{1}{14}x+\frac{2}{7}](https://tex.z-dn.net/?f=f%28x%29%3D14x-4%5C%5Cy%3D14x-4%5C%5Cx%3D14y-4%5C%5C14y%3Dx%2B4%5C%5Cy%3D%5Cfrac%7B1%7D%7B14%7Dx%2B%5Cfrac%7B2%7D%7B7%7D)
So
![f^{-1}(x)=\frac{1}{14}x+\frac{2}{7}](https://tex.z-dn.net/?f=f%5E%7B-1%7D%28x%29%3D%5Cfrac%7B1%7D%7B14%7Dx%2B%5Cfrac%7B2%7D%7B7%7D)
Domain: this is the range of the original so domain is (-∞,∞)
Range: this is the domain of the original so range is (-∞,∞)
6.
![f(x)=-3x+8](https://tex.z-dn.net/?f=f%28x%29%3D-3x%2B8)
Domain: There is no restriction on values of x we can put on it. Hence domain is (-∞,∞)
Range: no matter what we put into x, we can get any y value from negative infinity to positive infinity. So range is (-∞,∞)
<u>Finding the inverse:</u>
![f(x)=-3x+8\\y=-3x+8\\x=-3y+8\\3y=-x+8\\y=-\frac{1}{3}x+\frac{8}{3}](https://tex.z-dn.net/?f=f%28x%29%3D-3x%2B8%5C%5Cy%3D-3x%2B8%5C%5Cx%3D-3y%2B8%5C%5C3y%3D-x%2B8%5C%5Cy%3D-%5Cfrac%7B1%7D%7B3%7Dx%2B%5Cfrac%7B8%7D%7B3%7D)
So
![f^{-1}(x)=-\frac{1}{3}x+\frac{8}{3}](https://tex.z-dn.net/?f=f%5E%7B-1%7D%28x%29%3D-%5Cfrac%7B1%7D%7B3%7Dx%2B%5Cfrac%7B8%7D%7B3%7D)
Domain: this is the range of the original so domain is (-∞,∞)
Range: this is the domain of the original so range is (-∞,∞)