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Question 1209615: Suppose the domain of f is (-1,3). Define the function g by
g(x) = f(2/x + x/2).
What is the domain of g?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to determine the domain of g:
1. **Understand the relationship between f and g:**
The function g(x) is defined in terms of f. The input to f is the expression 2/x + x/2. For g(x) to be defined, this expression must fall within the domain of f, which is (-1, 3).
2. **Set up the inequality:**
We need to find the values of x for which:
-1 < 2/x + x/2 < 3
3. **Solve the inequalities:**
We'll solve this as two separate inequalities:
* **-1 < 2/x + x/2**
Multiply by 2x (assuming x is not 0, which we'll address later):
-2x < 4 + x²
0 < x² + 2x + 4
The discriminant of this quadratic is 2² - 4*1*4 = -12, which is negative. This means the quadratic is always positive, so this inequality is true for all x (except x=0).
* **2/x + x/2 < 3**
Multiply by 2x (again, assuming x is not 0):
4 + x² < 6x
x² - 6x + 4 < 0
Find the roots of x² - 6x + 4 = 0 using the quadratic formula:
x = (6 ± √(36 - 16))/2 = (6 ± √20)/2 = 3 ± √5
So, 3 - √5 < x < 3 + √5
4. **Combine the solutions and consider x=0:**
We need to find where *both* inequalities are true. Since the first inequality is true for all x (except 0) and the second inequality is true for 3 - √5 < x < 3 + √5, we simply need to consider the second inequality. Also, since we multiplied by x in the inequalities, we must exclude x=0 from the domain of g.
5. **State the domain of g:**
The domain of g is:
(3 - √5, 0) ∪ (0, 3 + √5)
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