CAT Functions and Domains Bible Top 10 Must Solve Functions and Domains Questions covering 90% of Syllabus – Page 2
All 39 Functions and Domains questions from CAT papers. Tap Show answer, or open the detailed solution (free login). Page 2 of 11.
For real values of x, the range of the function f(x) = (2x - 3)/(2x² + 4x - 6)
Answer: C. (-oo , 1/8] U [1/2 , oo)
Solution — first steps
Correct Answer: 3 We can factor the denominator: 2x² + 4x - 6 = 2(x + 3)(x - 1) So, the function becomes: f(x) = (2x - 3) / (2(x + 3)(x - 1)) , thus x ≠ -3 , 1 Range of f(x) f(0) = 1/2 f(1) = not possible f(2) = 1/10 f(3) = 1/8 Only option 3 includes f(0) and f(3) Thus option 3 is the answer
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f(x) = (x²+ 3x)(x² + 3x + 2), then the sum of all real roots of the equation √f(x) + 1 = 9701, is
Answer: C. -3
Solution — first steps
(a + 2) = a² + 2a √fx + 1 = √a² + 2a + 1 = 9701 √(a+1)² = 9701 , a = 9700 Thus x² + 3x = 9700 , x² + 3x - 9700 = 0 Sum of roots = -b/a = -3/1 = -3 Thus option 3 is the answer
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The sum of all real values of \(k\) for which \(\left(\frac{1}{8}\right)^k \times\left(\frac{1}{32768}\right)^{\frac{1}{3}}=\frac{1}{8} \times\left(\frac{1}{32768}\right)^{\frac{1}{k}}\), is
Answer: B. \(-\frac{2}{3}\)
Solution — first steps
Step 1. Convert every number to base 2: 1/8=2^-3 and 1/32768=2^-15.
Step 2. The left exponent of 2 is -3k-5; the right exponent is -3-15/k.
Step 2. The left exponent of 2 is -3k-5; the right exponent is -3-15/k.
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If the equations \(x^2+m x+9=0, x^2+n x+17=0\) and \(x^2+(m+n) x+35=0\) have a common negative root, then the value of \((2 m+3 n)\) is
Answer: 38
Solution — first steps
Step 1. Let the common negative root be r.
Step 2. Subtracting the sum of the first two equations from the third gives -r²+9=0.
Step 2. Subtracting the sum of the first two equations from the third gives -r²+9=0.
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All the values of \( x \) satisfying the inequality \( \frac{1}{x + 5} \leq \frac{1}{2x - 3} \) are:
Answer: D. \( x < -5 \) or \( \frac{3}{2} < x \leq 8 \)
Solution — first steps
Step 1: Try \( x = 0 \)
\[
\frac{1}{0 + 5} \leq \frac{1}{2(0) - 3} \Rightarrow \frac{1}{5} \leq \frac{1}{-3}
\Rightarrow 0.2 \leq -0.333 \quad \text{(False)}
\]
This means \( x = 0 \) does
not
satisfy the inequality.
- Option (1): \( -5 \frac{3}{2} \), so includes 100 ✅
- Option (4): only includes up to \( x \leq 8 \) ❌
So, eliminate Option 3.
✅ Final Answer: Option (4)
\[
x < -5 \quad \text{or} \quad \frac{3}{2} < x \leq 8
\]
\[
\frac{1}{0 + 5} \leq \frac{1}{2(0) - 3} \Rightarrow \frac{1}{5} \leq \frac{1}{-3}
\Rightarrow 0.2 \leq -0.333 \quad \text{(False)}
\]
This means \( x = 0 \) does
not
satisfy the inequality.
- Option (1): \( -5 \frac{3}{2} \), so includes 100 ✅
- Option (4): only includes up to \( x \leq 8 \) ❌
So, eliminate Option 3.
✅ Final Answer: Option (4)
\[
x < -5 \quad \text{or} \quad \frac{3}{2} < x \leq 8
\]
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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 14 repeating CAT PYQs and how much each adds
| CAT PYQ | Repeating CAT PYQ | Questions | Representative | Cumulative questions | Cumulative share |
|---|---|---|---|---|---|
| 1 | Reject nonphysical roots in quantity and time problems | 11 | CAT 2021 S1 Q16 | 11 | 21.15% |
| 2 | Logarithm base and argument restrictions | 10 | CAT 2021 S1 Q4 | 21 | 40.38% |
| 3 | Modulus branches and sign consistency | 6 | CAT 2021 S2 Q16 | 27 | 51.92% |
| 4 | Denominator zeros and composite or reciprocal inputs | 5 | CAT 2021 S1 Q1 | 32 | 61.54% |
| 5 | Radical arguments and square root signs | 4 | CAT 2023 S1 Q3 | 36 | 69.23% |
| 6 | Exponential substitutions zero bases and exponent conditions | 4 | CAT 2021 S2 Q15 | 40 | 76.92% |
| 7 | Feasible output range of rational expressions | 3 | CAT 2021 S2 Q13 | 43 | 82.69% |
| 8 | Piecewise function or region interval membership | 2 | CAT 2022 S3 Q6 | 45 | 86.54% |
| 9 | Positivity and nonzero roots in reciprocal identities | 2 | CAT 2023 S3 Q4 | 47 | 90.38% |
| 10 | Select a difference using its given sign | 1 | CAT 2021 S2 Q3 | 48 | 92.31% |
| 11 | Positive substitution needed for AM GM | 1 | CAT 2022 S3 Q2 | 49 | 94.23% |
| 12 | Given work time interval | 1 | CAT 2023 S3 Q8 | 50 | 96.15% |
| 13 | Select the negative shared root | 1 | CAT 2024 S1 Q20 | 51 | 98.08% |
| 14 | Open and closed floor band endpoints | 1 | CAT 2025 S1 Q22 | 52 | 100% |
Based on CAT Actual Quant Papers.