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Difficulty: HardFunction Evaluation, Composition, and Properties

Let the functions ff and gg be defined by f(x)=25−x2f(x) = \sqrt{25 - x^2} and g(x)=1x2−9g(x) = \frac{1}{\sqrt{x^2 - 9}} for all real numbers xx where the expressions are defined. What is the number of integers in the domain of the composite function h(x)=g(f(x))h(x) = g(f(x))?

Answer: 7

Answer

The number of integers in the domain of the composite function h(x)=g(f(x))h(x) = g(f(x)) is 7.
The composite function h(x)=g(f(x))h(x) = g(f(x)) is defined when xx is in the domain of ff and f(x)f(x) is in the domain of gg. The domain of f(x)=25−x2f(x) = \sqrt{25 - x^2} is [−5,5][-5, 5]. The domain of g(u)=1u2−9g(u) = \frac{1}{\sqrt{u^2 - 9}} is u>3u > 3 or u<−3u < -3. Substituting f(x)f(x) for uu gives 25−x2>3\sqrt{25 - x^2} > 3, which simplifies to x2<16x^2 < 16 or −4<x<4-4 < x < 4. The intersection of [−5,5][-5, 5] and (−4,4)(-4, 4) is (−4,4)(-4, 4). The integers in this interval are −3,−2,−1,0,1,2,3-3, -2, -1, 0, 1, 2, 3, which total 7 integers.

Step-by-Step Solution

1
Determine the domain of the inner function f(x)=25−x2f(x) = \sqrt{25 - x^2}.
The domain is [−5,5][-5, 5].
The term inside the square root must be greater than or equal to zero for the function to yield real values: 25−x2≥025 - x^2 \ge 0.
2
Determine the domain of the outer function g(u)=1u2−9g(u) = \frac{1}{\sqrt{u^2 - 9}}.
The domain is (−∞,−3)∪(3,∞)(-\infty, -3) \cup (3, \infty).
The expression inside the square root in the denominator must be strictly positive: u2−9>0u^2 - 9 > 0.
3
Apply the domain constraint of the outer function to the outputs of the inner function.
−4<x<4-4 < x < 4.
We require f(x)>3f(x) > 3 or f(x)<−3f(x) < -3. Since the range of f(x)f(x) is non-negative, f(x)<−3f(x) < -3 has no solutions. Thus, we solve 25−x2>3\sqrt{25 - x^2} > 3, which squares to 25−x2>925 - x^2 > 9, or x2<16x^2 < 16.
4
Find the intersection of the inner function's domain and the composite constraint.
The composite domain is (−4,4)(-4, 4).
The input xx must satisfy both the domain of ff (−5≤x≤5-5 \le x \le 5) and the composition constraint (−4<x<4-4 < x < 4).
5
List and count the integers within the composite domain (−4,4)(-4, 4).
There are 7 integers.
The integers strictly between −4-4 and 44 are −3,−2,−1,0,1,2,3-3, -2, -1, 0, 1, 2, 3.

Key Concept

Domain of a composite function
Estimated Time:2m 0s
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Topics

Function Evaluation, Composition, and Properties

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