13. Find the domain of [tex]f(x)=\sqrt{4-x^2}[/tex].
A. (-2,2)
B. [-2,2]
C. (-2,2)
D. [-2,2]
14. Find the domain of [tex]f(x)=\frac{5}{x^2-9}[/tex].
A. [tex]R \{9\}[/tex]
B. [tex]R \{-3,3\}[/tex]
C. [tex]R \{3\}[/tex]
D. [tex]R \{-9\}[/tex]
15. The domain of [tex]f(x)=x^2 \ln x[/tex] is
A. [tex](-\infty, \infty)[/tex]
B. [tex](1, \infty)[/tex]
C. [tex](0,1)[/tex]
D. [tex](0, \infty)[/tex].
16. Find the domain of [tex]\sqrt{4-7 t}[/tex].
A. [tex][0, \infty)[/tex]
B. [tex]\frac{7}{4}[/tex]
C. [tex](-\infty, \frac{7}{4}][/tex]
D. [tex](\frac{7}{4}, \infty)[/tex].
17. Let [tex]f: R \rightarrow R[/tex] be defined by [tex]f(x)=\frac{5}{x^2-1}[/tex]. Find the domain of [tex]f[/tex].
A. [tex]R \{4\}[/tex]
B. [tex]R \{2\}[/tex]
C. [tex]R \{-4\}[/tex]
D. [tex]R \{-2,2\}[/tex].
18. Find the inverse [tex]g^{-1}(x)[/tex] of [tex]g(x)=\sqrt{x-3}[/tex]
A. [tex](x-3)^2[/tex]
B. [tex]x^2+3[/tex]
C. [tex]x^2-3[/tex]
D. [tex]\frac{1}{x-3}[/tex].
Answer (1)
Find the domain of f ( x ) = 4 − x 2 .
Set up the inequality 4 − x 2 ≥ 0 .
Solve the inequality to find − 2 ≤ x ≤ 2 .
The domain is [ − 2 , 2 ] . [ − 2 , 2 ]
Explanation
Analyze the problem We need to find the domain of the function f ( x ) = 4 − x 2 . The domain consists of all real numbers x for which the expression inside the square root is non-negative. Therefore, we need to solve the inequality 4 − x 2 ≥ 0 .
Solve the inequality We can rewrite the inequality as x 2 ≤ 4 . Taking the square root of both sides, we get ∣ x ∣ ≤ 2 , which means − 2 ≤ x ≤ 2 .
Determine the domain Therefore, the domain of f ( x ) is the interval [ − 2 , 2 ] .
Final Answer The correct answer is D. [ − 2 , 2 ] .
Examples
Understanding the domain of a function is crucial in many real-world applications. For example, if f ( x ) represents the height of an arch at a distance x from its center, the domain [ − 2 , 2 ] tells us the physical boundaries of the arch. Similarly, if f ( x ) represents the population of a species x years from now, a domain restriction might indicate a time frame over which the model is valid. In general, determining the domain helps us understand the limitations and applicability of mathematical models in various contexts.
