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Question 1209612: Suppose the domain of f is (-1,1). Define the function g by
g(x) = 5 - f(x) + f(5/x).
What is the domain of g?
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
f(5/x) is what we should focus on.
The input 5/x needs to be on the interval -1 < 5/x < 1, so it fits with the domain interval (-1,1).
You should find that -1 < 5/x < 1 leads to either x < -5 or x > 5.
But none of those values are in the interval -1 < x < 1.
Therefore the domain of g(x) is the empty set.
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Some examples:
Let's try x = 0.5
g(x) = 5 - f(x) + f(5/x)
g(0.5) = 5 - f(0.5) + f(5/0.5)
g(0.5) = 5 - f(0.5) + f(10)
g(0.5) = 5 - f(0.5) + undefined
g(0.5) = undefined
This shows that x = 0.5 is not in the domain of g(x).
f(10) is undefined since f(x) is only defined when -1 < x < 1.
Let's try x = 10
g(x) = 5 - f(x) + f(5/x)
g(10) = 5 - f(10) + f(5/10)
g(10) = 5 - f(10) + f(0.5)
g(10) = 5 - undefined + f(0.5)
g(10) = undefined
We run into the same problem as before.
This shows that x = 10 is not in the domain of g(x).
I encourage the student to explore other examples.
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