Answer:
1.
<u>Function:</u>
Domain: (-∞,∞)
Range: (-∞,0]
<u>Inverse Function:</u>
Domain: (-∞,0]
Range: (-∞,∞)
2.
<u>Function:</u>
Domain: (-∞,∞)
Range: (-∞,∞)
<u>Inverse Function:</u>
Domain: (-∞,∞)
Range: (-∞,∞)
3.
<u>Function:</u>
Domain: (-∞,∞)
Range: (-∞,∞)
<u>Inverse Function:</u>
Domain: (-∞,∞)
Range: (-∞,∞)
4.
<u>Function:</u>
Domain: (-∞,∞)
Range: [7,∞)
<u>Inverse Function:</u>
Domain: [7,∞)
Range: (-∞,∞)
5.
<u>Function:</u>
Domain: (-∞,∞)
Range: (-∞,∞)
<u>Inverse Function:</u>
Domain: (-∞,∞)
Range: (-∞,∞)
6.
<u>Function:</u>
Domain: (-∞,∞)
Range: (-∞,∞)
<u>Inverse Function:</u>
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,
<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.
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>
So
Domain: this is the range of the original so domain is (-∞,0]
Range: this is the domain of the original so range is (-∞,∞)
2.
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>
So
Domain: this is the range of the original so domain is (-∞,∞)
Range: this is the domain of the original so range is (-∞,∞)
3.
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>
So
Domain: this is the range of the original so domain is (-∞,∞)
Range: this is the domain of the original so range is (-∞,∞)
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, 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>
So
Domain: this is the range of the original so domain is [7,∞)
Range: this is the domain of the original so range is (-∞,∞)
5.
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>
So
Domain: this is the range of the original so domain is (-∞,∞)
Range: this is the domain of the original so range is (-∞,∞)
6.
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>
So
Domain: this is the range of the original so domain is (-∞,∞)
Range: this is the domain of the original so range is (-∞,∞)