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Functions and Graphs questions for IPMAT, with solutions

10 free Functions and Graphs practice questions for IPMAT Indore, Rohtak and JIPMAT, 4 from past papers, each with the answer and a full worked solution. No sign-in. Try each one first, then open the solution. The lessons for this chapter are in Functions and Graphs on Learn.

1. Question 1Multiple choiceAsked in IPMAT Indore 2020Functions

   Given $f(x)=x^{2}+\log_{3}x$ and $g(y)=2y+f(y)$, then the value of $g(3)$ equals

   1. A.$16$
   2. B.$15$
   3. C.$25$
   4. D.$26$
   Show the answer and solution

   Answer: A. $16$

   First evaluate $f(3)$: $f(3)=3^2+\log_33=9+1=10$. Then $g(3)=2\cdot3+f(3)=6+10=16$. The common slip is forgetting that $\log_33=1$, which gives $15$.

2. Question 2Multiple choiceWritten by IPMATwayFunctions

   If $f$ is an odd function with $f(3)=5$, then $f(-3)+f(0)$ equals

   1. A.$0$
   2. B.$-4$
   3. C.$-5$
   4. D.$5$
   Show the answer and solution

   Answer: C. $-5$

   An odd function satisfies $f(-x)=-f(x)$. So $f(-3)=-f(3)=-5$. Putting $x=0$: $f(0)=-f(0)$, so $f(0)=0$. Hence $f(-3)+f(0)=-5+0=-5$.

3. Question 3Type the answerWritten by IPMATwayFunctions

   For real numbers $a$ and $b$, define $a@b=a^{2}+b^{2}$ and $a\#b=a-b$. Then $(3@4)\#(2@1)$ equals

   Show the answer and solution

   Answer: 20

   Evaluate each bracket with $@$: $3@4=3^2+4^2=25$ and $2@1=2^2+1^2=5$. Then apply $\#$: $25\#5=25-5=20$.

4. Question 4Multiple choiceAsked in IPMAT Indore 2025Graphs and Behaviour

   If $y=a+b\log_{e}x$, then

   1. A.$y-a$ is proportional to $x^{b}$
   2. B.$\log_{e}y$ is proportional to $x$
   3. C.$\dfrac{1}{y-a}$ is proportional to $x^{b}$
   4. D.$e^{y}$ is proportional to $x^{b}$
   Show the answer and solution

   Answer: D. $e^{y}$ is proportional to $x^{b}$

   Exponentiate both sides: $e^y=e^{a+b\log_ex}=e^a\cdot e^{b\log_ex}$. Since $e^{b\log_ex}=\left(e^{\log_ex}\right)^b=x^b$, we get $e^y=e^a\,x^b$. As $e^a$ is a constant, $e^y$ is proportional to $x^b$.

5. Question 5Multiple choiceAsked in IPMAT Rohtak 2019Graphs and Behaviour

   If the minimum value of $f(x)=x^{2}+2bx+2c^{2}$ is greater than the maximum value of $g(x)=-x^{2}-2cx+b^{2}$ for real $x$, then

   1. A.$|c|>\sqrt{2}|b|$
   2. B.$\sqrt{2}|c|>b$
   3. C.$0<c<\sqrt{2}b$
   4. D.no real values of $b$ and $c$
   Show the answer and solution

   Answer: A. $|c|>\sqrt{2}|b|$

   Complete the squares: $f(x)=(x+b)^2+2c^2-b^2$, so $\min f=2c^2-b^2$. And $g(x)=-(x+c)^2+b^2+c^2$, so $\max g=b^2+c^2$. The condition $2c^2-b^2>b^2+c^2$ simplifies to $c^2>2b^2$, i.e. $|c|>\sqrt2\,|b|$.

6. Question 6Type the answerWritten by IPMATwayRelations

   The number of onto functions from a set with $4$ elements to a set with $2$ elements is

   Show the answer and solution

   Answer: 14

   There are $2^4=16$ functions from a $4$-element set to a $2$-element set. A function fails to be onto only if it misses one of the two values, i.e. it is constant. There are $2$ constant functions. So the number of onto functions is $16-2=14$.

7. Question 7Type the answerWritten by IPMATwayRelations

   The maximum number of equivalence relations on the set $A=\{1,2,3\}$ is

   Show the answer and solution

   Answer: 5

   Each equivalence relation on $A$ corresponds to a partition of $A$ into classes. The partitions of $\{1,2,3\}$ are: $\{1\},\{2\},\{3\}$; $\{1,2\},\{3\}$; $\{1,3\},\{2\}$; $\{2,3\},\{1\}$; and $\{1,2,3\}$. That is $5$ partitions, so there are $5$ equivalence relations.

8. Question 8Multiple choiceAsked in IPMAT Indore 2021Functions

   Suppose that a real-valued function $f(x)$ of real numbers satisfies $f(x+xy)=f(x)+f(xy)$ for all real $x$, $y$, and that $f(2020)=1$. Compute $f(2021)$.

   1. A.$\dfrac{2021}{2020}$
   2. B.$\dfrac{2020}{2019}$
   3. C.$1$
   4. D.$\dfrac{2020}{2021}$
   Show the answer and solution

   Answer: A. $\dfrac{2021}{2020}$

   Additivity gives $f(nx)=nf(x)$ for integers and $f\left(\tfrac{p}{q}x\right)=\tfrac{p}{q}f(x)$. With $r=\tfrac{2021}{2020}$ and $x=2020$: $f(2021)=\tfrac{2021}{2020}$.

9. Question 9Multiple choiceWritten by IPMATwayGraphs and Behaviour

   How many real values of $x$ satisfy $|x^2-4x|=4$?

   1. A.$4$
   2. B.$2$
   3. C.$1$
   4. D.$3$
   Show the answer and solution

   Answer: D. $3$

   $|x^2-4x|=4$ means $x^2-4x=4$ or $x^2-4x=-4$. First: $x^2-4x-4=0$ has discriminant $16+16=32>0$, roots $x=2\pm2\sqrt2$, two distinct values. Second: $x^2-4x+4=0$, i.e. $(x-2)^2=0$, one repeated root $x=2$. This differs from $2\pm2\sqrt2$. In the graph, the line $y=4$ cuts the W-shaped curve at two outer points and touches the middle peak at $(2,4)$. Total of $3$ solutions.

10. Question 10Multiple choiceWritten by IPMATwayRelations

    On the set $\{1,2,3,4\}$, how many relations are reflexive and symmetric but not transitive?

    1. A.$49$
    2. B.$64$
    3. C.$15$
    4. D.$63$
    Show the answer and solution

    Answer: A. $49$

    A reflexive relation must contain $(i,i)$ for all four elements. A symmetric relation includes $(i,j)$ together with $(j,i)$, so each of the $\binom42=6$ unordered pairs is in or out: $2^6=64$ relations. Those that are also transitive are exactly the equivalence relations, which correspond to partitions of a 4-element set: Bell number $B_4=15$. Hence reflexive, symmetric, not transitive: $64-15=49$.