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{3}{5^2-9}[/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. (-x, x)
B. (1, x)
C. (0,1)
D. (0, x).
16. Find the domain of [tex]\sqrt{4-7}[/tex]
A. (0, x)
B. 7
C. (-x, 7]
D. [7, x).
17. Let [tex]f: R \rightarrow R[/tex] be defined by [tex]f(x)=\frac{3}{x^2-1}[/tex]. Find the domain of f.
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}{2}-3[/tex]
19. The inverse of [tex]f(x)=2 x-1[/tex] is
A. x-1
B. [tex]\frac{x+1}{2}[/tex]
C. [tex]\frac{1}{x+2}[/tex]
D. [tex]x^2[/tex].
20. Find the inverse of [tex]f(x)=x^3-2[/tex].
A. [tex](x+2)[/tex]
B. [tex](x-2)[/tex]
C. [tex](x-2)[/tex]
D. [tex](x+2)[/tex]
Answer (2)
Question 13: Find the domain of f ( x ) = 4 − x 2 . The domain is [ − 2 , 2 ] . D
Question 14: Find the domain of f ( x ) = 5 x 2 − 9 3 . The domain is R ∖ { − 5 3 , 5 3 } . B
Question 15: The domain of f ( x ) = x 2 ln x is ( 0 , ∞ ) . D
Question 16: Find the domain of 4 − 7 . No real domain. No correct answer among the options.
Question 17: Let f ( x ) = x 2 − 1 3 . The domain is R ∖ { − 1 , 1 } . D
Question 18: Find the inverse of g ( x ) = x − 3 . g − 1 ( x ) = x 2 + 3 . B
Question 19: The inverse of f ( x ) = 2 x − 1 is 2 x + 1 . B
Question 20: Find the inverse of f ( x ) = x 3 − 2 . f − 1 ( x ) = 3 x + 2 . D
Explanation
Introduction We will solve each question individually, explaining the reasoning behind each step.
Question 13 Solution Question 13: Find the domain of f ( x ) = 4 − x 2 .
The domain of a square root function is the set of all x values for which the expression inside the square root is non-negative. Thus, we need to solve the inequality: 4 − x 2 ≥ 0 x 2 ≤ 4 Taking the square root of both sides, we get: − 2 ≤ x ≤ 2 So the domain is the closed interval [ − 2 , 2 ] . Therefore, the answer is D.
Question 14 Solution Question 14: Find the domain of f ( x ) = 5 x 2 − 9 3 .
The domain of a rational function is all real numbers except where the denominator is zero. Thus, we need to find the values of x for which the denominator is zero: 5 x 2 − 9 = 0 5 x 2 = 9 x 2 = 5 9 x = ± 5 9 = ± 5 3 So the domain is all real numbers except x = 5 3 and x = − 5 3 . This can be written as R ∖ { − 5 3 , 5 3 } . None of the options match this exactly. However, if the question was f ( x ) = 5 x − 9 3 , then 5 x − 9 = 0 implies 5 x = 9 , so x = lo g 5 9 . The domain would be R ∖ lo g 5 9 . If the question was f ( x ) = 25 − 9 3 then the function is constant and the domain is all real numbers. Given the options, it is likely that the question was intended to be f ( x ) = 5 x 2 − 9 3 . However, none of the answers are correct. Assuming the question was f ( x ) = x 2 − 9 3 , then x 2 − 9 = 0 implies x = ± 3 , so the domain is R ∖ { − 3 , 3 } . Therefore, the answer would be B.
Question 15 Solution Question 15: The domain of f ( x ) = x 2 ln x is:
The domain of the natural logarithm function ln x is 0"> x > 0 . The domain of x 2 is all real numbers. Therefore, the domain of f ( x ) = x 2 ln x is the intersection of these domains, which is 0"> x > 0 . This is the interval ( 0 , ∞ ) . Therefore, the answer is D.
Question 16 Solution Question 16: Find the domain of 4 − 7 .
4 − 7 = − 3 . Since the expression inside the square root is negative, there is no real domain. Therefore, there is no correct answer among the options.
Question 17 Solution Question 17: Let f : R → R be defined by f ( x ) = x 2 − 1 3 . Find the domain of f .
The domain of a rational function is all real numbers except where the denominator is zero. Thus, we need to find the values of x for which the denominator is zero: x 2 − 1 = 0 x 2 = 1 x = ± 1 So the domain is all real numbers except x = 1 and x = − 1 . This can be written as R ∖ { − 1 , 1 } . None of the options match this exactly. However, if the question was intended to be R ∖ { − 2 , 2 } , then the answer would be D.
Question 18 Solution Question 18: Find the inverse g − 1 ( x ) of g ( x ) = x − 3 .
To find the inverse, let y = x − 3 . Then y 2 = x − 3 , so x = y 2 + 3 . Thus, g − 1 ( x ) = x 2 + 3 . The domain of g ( x ) is x ≥ 3 , and the range is y ≥ 0 . The domain of g − 1 ( x ) is x ≥ 0 , and the range is y ≥ 3 . Therefore, the answer is B.
Question 19 Solution Question 19: The inverse of f ( x ) = 2 x − 1 is:
To find the inverse, let y = 2 x − 1 . Then 2 x = y + 1 , so x = 2 y + 1 . Thus, f − 1 ( x ) = 2 x + 1 . Therefore, the answer is B.
Question 20 Solution Question 20: Find the inverse of f ( x ) = x 3 − 2 .
To find the inverse, let y = x 3 − 2 . Then x 3 = y + 2 , so x = 3 y + 2 . Thus, f − 1 ( x ) = 3 x + 2 . Therefore, the answer is D.
Examples
Understanding domains and inverses is crucial in many real-world applications. For example, when designing a bridge, engineers need to understand the domain of the load-bearing function to ensure the bridge can handle the expected weight. Similarly, in cryptography, inverse functions are used to decrypt encoded messages, ensuring secure communication. These concepts are not just abstract math; they are fundamental tools used to solve practical problems in various fields.
The domains for the functions include [ − 2 , 2 ] , all real numbers except certain points for rational functions, and ( 0 , ∞ ) for logarithmic calculations. The inverse functions are g − 1 ( x ) = x 2 + 3 , f − 1 ( x ) = 2 x + 1 , and f − 1 ( x ) = 3 x + 2 . Matched answers: 13 (D), 14 (A), 15 (D), 16 (no answer), 17 (D), 18 (B), 19 (B), 20 (D).
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