The sum of two numbers is 45. One number is 4 times as large as the other. What are the numbers?
1 answer:
We can define two variables:
x = large number
y = small number
Now, set up a system of equations:
Solve the system. I'm going to use the second equation to substitute 'x' into the first one (x = 4y):
Plug that back into 2nd equation to find both numbers:
So, x = 36 and y = 9.
We can check. 36 + 9 actually does equal 45 and 4 times 9 is 36. We match our initial criteria, so our answer is right.
x = large number
y = small number
Now, set up a system of equations:
Solve the system. I'm going to use the second equation to substitute 'x' into the first one (x = 4y):
Plug that back into 2nd equation to find both numbers:
So, x = 36 and y = 9.
We can check. 36 + 9 actually does equal 45 and 4 times 9 is 36. We match our initial criteria, so our answer is right.
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<h3>Answer: -7 < x < 17</h3>
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Explanation:
Plug in the lower bound of the domain, which is x = -3
f(x) = 3x+2
f(-3) = 3(-3)+2
f(-3) = -9+2
f(-3) = -7
If x = -3, then the output is y = -7. Since f(x) is an increasing function (due to the positive slope), we know that y = -7 is the lower bound of the range.
If you plugged in x = 5, you should find that f(5) = 17 making this the upper bound of the range.
The range of f(x) is -7 < y < 17
Recall that the domain and range swap places when going from the original function f(x) to the inverse
This swap happens because how x and y change places when determining the inverse itself. In other words, you go from y = 3x+2 to x = 3y+2. Solving for y gets us y = (x-2)/3 which is the inverse.
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In short, we found the range of f(x) is -7 < y < 17.
That means the domain of the inverse is -7 < x < 17 since the domain and range swap roles when going from original to inverse.