For all real values of x, the range of the function \(f(x) = \dfrac{x^2 + 2x + 4}{2x^2 + 4x + 9}\) is: Options: A) \([\dfrac{4}{9}, \dfrac{8}{9}]\) B) \([\dfrac{3}{7}, \dfrac{8}{9})\) C) \((\dfrac{3}{7}, \dfrac{1}{2})\) D) \([\dfrac{3}{7}, \dfrac{1}{2})\)

D. \([\dfrac{3}{7}, \dfrac{1}{2})\). Step 1. Put t=(x+1)²≥0. Step 2. Function becomes(t+3)/(2t+7)=1/2−1/[2(2t+7)].

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CAT 2021 Slot 2 · Quant

Refer to the passage above ↑

For all real values of x, the range of the function \(f(x) = \dfrac{x^2 + 2x + 4}{2x^2 + 4x + 9}\) is:

Last day solving checklist

Name the unknown. Include its unit and state whether it is a count, amount, rate or parameter.
Form the relationships. Translate each given condition into an equation or restriction.
Choose the method. Use the formula, substitution, elimination, a sign chart or a bound appropriate to this concept.
Check restrictions. Check integer counts, signs, nonzero denominators, domains and all original conditions.
Verify the answer. Substitute into the original question; for an extreme, show a feasible example reaches it.

All 21 repeating CAT PYQs and how much each adds

CAT PYQ	Repeating CAT PYQ	Questions	Representative	Cumulative questions	Cumulative share
1	Exponential bounds and smallest integer periods	7	CAT 2021 S2 Q15	7	12.73%
2	Discriminant bounds for real roots or ranges	6	CAT 2022 S1 Q5	13	23.64%
3	Linear allocation bounds with nonnegative counts	6	CAT 2022 S1 Q8	19	34.55%
4	Absolute value intervals and sign cases	5	CAT 2021 S1 Q2	24	43.64%
5	Ordered values and average bounds	5	CAT 2022 S2 Q8	29	52.73%
6	Logarithmic bounds and monotonicity	4	CAT 2024 S2 Q9	33	60.00%
7	Rational sign charts and excluded poles	3	CAT 2021 S1 Q1	36	65.45%
8	Set overlap bounds and feasible endpoints	3	CAT 2022 S1 Q17	39	70.91%
9	Finite integer enumeration within bounds	3	CAT 2025 S1 Q17	42	76.36%
10	Floor intervals and endpoint inclusion	2	CAT 2022 S1 Q12	44	80.00%
11	Factorial size bounds	1	CAT 2022 S2 Q13	45	81.82%
12	Bounds on eligible AP terms	1	CAT 2022 S3 Q17	46	83.64%
13	Remainders with an upper population bound	1	CAT 2022 S3 Q19	47	85.45%
14	Polygon side inequality	1	CAT 2022 S3 Q22	48	87.27%
15	Speed bound with a palindrome restriction	1	CAT 2023 S1 Q8	49	89.09%
16	Translate time bounds to work rate bounds	1	CAT 2023 S3 Q8	50	90.91%
17	Quadratic percentage growth inequality	1	CAT 2023 S3 Q10	51	92.73%
18	Geometric regions defined by inequalities	1	CAT 2024 S1 Q4	52	94.55%
19	Square bound with an equality condition	1	CAT 2024 S1 Q8	53	96.36%
20	Compare quadratic extrema	1	CAT 2025 S1 Q8	54	98.18%
21	Range by completing the square and reciprocal bounds	1	CAT 2021 S2 Q13	55	100%

Based on CAT Actual Quant Papers.