Algebra: Functions Questions - 2 | Quantitative Aptitude (Quant) - CAT PDF Download

Question 12: The value of f∘g∘h(9) could be, if

f(x) =1/x
g(x) = 1/(x-2)
h(x) = √x 

A. 3
B. 1/3
C. -5
D. None of these

Ans: (D)

Explanation: Evaluate from the innermost function outwards. $h(9)=\sqrt{9}=3$ (principal square root is non-negative). Then $g(3)=\dfrac{1}{3-2}=1$. Finally $f(1)=\dfrac{1}{1}=1$. The final value is $1$, which does not match options A, B or C. Therefore the correct choice is D (None of these).

Question 13: For this question, assume the following operators: 
A*B = A² - B² 

A-B = A/B

A+B = A * B 

A/B = A+B 

Which of the following expression would yield the result as x subtracted by y?

A. (x*y)-(x+5)
B. (x/y)*(x-y)
C. (x*y)-(x/y)
D. (x+y)*(x-y)

Ans: (C)

Sol: Use the operator definitions and interpret the outer operators correctly.

x*y = x² - y², and x/y = x + y.

Compute option (C): (x*y) - (x/y).

Remember that here '-' is defined by A - B = A/B, so (x*y) - (x/y) means

$(x*y) / (x/y) = \dfrac{x^{2} - y^{2}}{\,x + y\,}.$

Simplify: $\dfrac{x^{2} - y^{2}}{x + y} = \dfrac{(x - y)(x + y)}{x + y} = x - y$, provided $x + y \neq 0$.

Hence option (C) yields $x - y$.

Question 14: Find the domain of:

A. (-∞,9)
B. [-1,9)
C. [-1,9) excluding 0
D. (-1,9)

Ans: (D)

Sol: Determine permissible x from each restriction in the expression.

1. The logarithm argument must be positive: $9 - x > 0 \Rightarrow x < 9$.

2. Any denominator of the form $1 - \log(9 - x)$ must not be zero, so $\log(9 - x) \neq 1 \Rightarrow 9 - x \neq 10 \Rightarrow x \neq -1$.

3. If a factor $x + 1$ appears where positivity is required, that would give $x > -1$; combining this with the exclusion $x \neq -1$ keeps $x > -1$.

Combining $x > -1$ and $x < 9$ gives the domain $(-1, 9)$.

Question 15: If [X] - Greatest integer less than or equal to x. Find the value of 
[√1] + [√2] + [√3] +............................................................+ [√100]

A. 615
B. 625
C. 5050
D. 505

Ans: (B)

Sol: For $k=1,2,\dots,9$, the integers $n$ with $\lfloor\sqrt{n}\rfloor=k$ run from $k^{2}$ to $(k+1)^{2}-1$, a total of $2k+1$ numbers. Each such $n$ contributes $k$ to the sum, so the contribution is $k(2k+1)$.

Additionally, $n=100$ gives $\lfloor\sqrt{100}\rfloor=10$, which contributes $10$.

Total $$\;=\sum_{k=1}^{9} k(2k+1) + 10 \;=\; 2\sum_{k=1}^{9}k^{2} + \sum_{k=1}^{9}k + 10.$$

Using $\sum_{k=1}^{9}k = 45$ and $\sum_{k=1}^{9}k^{2} = 285$,

Total $= 2\times285 + 45 + 10 = 570 + 45 + 10 = 625.$

Question 16: Find the value of x for which x[x] = 39?

A. 6.244
B. 6.2
C. 6.3
D. 6.5

Ans: (D)

Sol: Let $n=\lfloor x\rfloor$ (an integer). Then $x\cdot n = 39$, so $x = 39/n$, and this $x$ must satisfy $n \le x < n+1$.

Try $n=6$: $x = 39/6 = 6.5$, and $6 \le 6.5 < 7$, so this is valid.

Other integer choices for $n$ do not yield $x$ in the required interval. Thus $x = 6.5$.

Question 17: Find the value of x for which x[x] = 15?

A. 3.5
B. 5
C. 6.1
D. None of these

Ans: (D)

Sol: Let $n=\lfloor x\rfloor$. Then $x = 15/n$ and must satisfy $n \le x < n+1$.

Test integer $n$ values:

$n=3\Rightarrow x=15/3=5$, but $5 \not\in [3,4)$, so inconsistent.

$n=4\Rightarrow x=15/4=3.75$, but $3.75 \not\in [4,5)$, inconsistent.

$n=5\Rightarrow x=15/5=3$, but $3 \not\in [5,6)$, inconsistent.

Similar checks for other integers fail. Hence there is no real $x$ satisfying $x\lfloor x\rfloor = 15$, so the answer is None of these.

Question 18: If f(x) =1/g(x), then which of the following is correct?

A. f(f(g(g(f(x))))) = g(f(g(g(g(x)))))
B. f(f(f(g(g(g(f(g(x)))))))) = g(g(g(g(f(g(f(f(x))))))))
C. f(f(g(f(x)))) = g(g(f(g(x))))
D. f(g(f(f(g(f(g(g(x)))))))) = g(g(g(g(f(f(f(f(x))))))))

Ans: (D)

Explanation: From $f(x)=1/g(x)$ we have the multiplicative relation $f(x)\,g(x)=1$. A necessary condition for the two sides of a composition identity to be potentially equal for all $x$ is that the total numbers of $f$ and $g$ applications are the same on each side.

Count the occurrences of $f$ and $g$ in each option. Only option (D) has four $f$'s and four $g$'s on each side (every other option has unequal counts). Therefore (D) is the only choice that can match in the required sense, so (D) is correct.

Question 19: If f(x) =

Find the value of x for which f(x) = f^-1(x)?

A. -3
B. 2
C. Both A and B
D. None of these

Ans: (C)

Sol: $f(x)=f^{-1}(x)$ implies $f(x)=x$. Using the given function (as shown in the image), this reduces to

$x^{2}+x-6=0 \Rightarrow (x+3)(x-2)=0.$

Thus $x=-3$ or $x=2$. Both values satisfy $f(x)=f^{-1}(x)$, so option C is correct.

Question 20: If f(x) = |x| + |x+3| + |x+6| + ...................................+ |x+3t|, where x is an integer and t is a positive integer, find the minimum value of f(x) when t = 6?

A. 63
B. 36
C. 30
D. 25

Ans: (B)

Sol: For $t=6$ there are $7$ terms: $|x|,\;|x+3|,\;|x+6|,\;|x+9|,\;|x+12|,\;|x+15|,\;|x+18|$. The sum of absolute values is minimised when $x$ is the median of the points $\{0,-3,-6,-9,-12,-15,-18\}$.

The median is $-9$. Evaluate $f(-9)$:

$|{-9}| + |{-6}| + |{-3}| + |0| + |3| + |6| + |9| = 9 + 6 + 3 + 0 + 3 + 6 + 9 = 36.$

Therefore the minimum value is $36$.

Question 21: In the previous question if t = 7, for how many values of x, f(x) will be minimum?

A. 1
B. 2
C. 4
D. 8

Ans: (C)

Sol: For $t=7$ there are $8$ terms: $|x|, |x+3|, \dots, |x+21|$, corresponding to the points $0,-3,-6,\dots,-21$. The two middle points (4th and 5th) of this sorted list are $-12$ and $-9$.

The sum of absolute deviations is minimised for any $x$ in the interval $[-12,-9]$. For integer $x$ this gives $x=-12,-11,-10,-9$, which are $4$ values. Hence option (C).

Question 22: If f(x² - 1) = x⁴ - 7x² + k1 and f(x³ - 2) = x⁶ - 9x³ + k₂ then the value of (k₂ - k₁) is

A. 6
B. 7
C. 8
D. 9
E. None of the above

Ans: (C)

Approach and Solution

Set $x$ so that the function argument becomes $0$ and equate the two expressions for $f(0)$.

If $x^{2}=1$ (take $x=\pm1$) then $f(0)=1^{2}-7\cdot1+k_{1} = -6 + k_{1}$. (1)

If $x^{3}=2$ then $x^{6}=4$, so $f(0)=4 - 9\cdot2 + k_{2} = 4 - 18 + k_{2} = -14 + k_{2}$. (2)

Equate (1) and (2): $-6 + k_{1} = -14 + k_{2} \Rightarrow k_{2} - k_{1} = 8.$

Question 23: Which of the following is not an odd function?

A. f(x) = -x³
B. f(x) = x⁵
C. f(x) = x² - x
D. f(x) = |x|³

Ans: (C) and (D)

Explanation: A function is odd if $f(-x) = -f(x)$ for all $x$.

A. For $f(x) = -x^{3}$, $f(-x) = -(-x)^{3} = x^{3} = -f(x)$, so it is odd.

B. For $f(x) = x^{5}$, $f(-x) = (-x)^{5} = -x^{5} = -f(x)$, so it is odd.

C. For $f(x) = x^{2} - x$, $f(-x) = x^{2} + x \neq -(x^{2} - x) = -x^{2} + x$, so it is not odd.

D. For $f(x) = |x|^{3}$, $f(-x) = |{-x}|^{3} = |x|^{3} = f(x)$, so it is even (hence not odd).