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Question 1209613: Suppose the domain of f is (-1,3). Define the function g by
g(x) = f(x - 4x^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 x - 4x². 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 < x - 4x² < 3
3. **Solve the inequalities:**
We'll solve this as two separate inequalities:
* **-1 < x - 4x²**
4x² - x - 1 < 0
Find the roots of 4x² - x - 1 = 0 using the quadratic formula:
x = (1 ± √(1 + 16))/8 = (1 ± √17)/8
So, (1 - √17)/8 < x < (1 + √17)/8
* **x - 4x² < 3**
4x² - x + 3 > 0
Find the roots of 4x² - x + 3 = 0. The discriminant is 1 - 4*4*3 = -47, which is negative. This means the quadratic is always positive, so this inequality is true for all x.
4. **Combine the solutions:**
Since the second inequality is true for all x, we only need to consider the first inequality.
5. **State the domain of g:**
The domain of g is:
((1 - √17)/8, (1 + √17)/8)
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