Explanation
To find the interval where h(x)=f(g(x))=1, we look at the definition of f(u). From the problem, f(u)=1 when ∣u∣≤1. Therefore, h(x)=1 when ∣g(x)∣≤1.
1. Analyzing the condition ∣g(x)∣≤1:
Case 1: ∣x∣≤2
In this range, g(x)=2−x2.
The condition becomes ∣2−x2∣≤1, which is equivalent to:
−1≤2−x2≤1
Subtracting 2 from all parts:
−3≤−x2≤−1
Multiplying by −1 (and reversing the inequalities):
1≤x2≤3
Taking the square root:
1≤∣x∣≤3
Case 2: |x| > 2
In this range, g(x)=2.
The condition becomes ∣g(x)∣≤1⟹∣2∣≤1, which is false.
2. Conclusion:
Combining the cases, h(x)=1 when 1≤∣x∣≤3.
Comparing this to the given options, we find that the correct choice is (d).
Correct Option: (d)
