Question 1 of 78
ThinkCAT Practice SetQAAlgebra • FunctionsEasy
If and $h(x) = x^2 - 1 , what is $g(h(2))$?
Official Correct Answer: A. First, calculate $h(2) = 2^2 - 1 = 3$. Then, $g(h(2)) = g(3) = 2(3) + 3 = 9$. Thus, the correct answer is A.
Algebra • Functions & Graphs Practice Drills (78 Qs)• Official Answer Keys
CAT Functions & Graphs Practice Questions (75+ Questions)
Domain and Range, Composite Functions, Even-Odd Functions, Functional Equations, and Graph Transformations
78 Total Questions
MCQ: 59 (+3 / -1)
TITA: 19 (0 Negative Penalty)
Core Formulas & Shortcut Matrix: Functions & Graphs Practice Drills
Composite FunctionFormula #1
(f \circ g)(x) = f(g(x))
Evaluate inner function g(x) first before applying f.
Cauchy Functional EquationFormula #2
f(x+y) = f(x)f(y) \implies f(x) = a^x
Standard functional form testing exponential growth.
Additive Functional EquationFormula #3
f(xy) = f(x) + f(y) \implies f(x) = k \ln x
Standard logarithmic functional form.
Exam Hall Traps & Speedbreakers to Avoid
- •Finding values outside the domain of the inner function in composite functions.
Bite-Sized Practice Sets (9 Modules Available)
Authentic CAT BlueprintSolve in structured 20–25 question practice sets with instant accuracy benchmarking and solution checks. Sets 01 & 02 are completely free.
12 QuestionsPro Pass
Set 01: Functions
Difficulty: Easy
8 QuestionsPro Pass
Set 02: Functions
Difficulty: Hard
13 QuestionsPro Pass
Set 03: Functions
Difficulty: Hard
9 QuestionsPro Pass
Set 04: Functions
Difficulty: Hard
8 QuestionsPro Pass
Set 05: Functions
Difficulty: Easy
5 QuestionsPro Pass
Set 06: Functions
Difficulty: Hard
9 QuestionsPro Pass
Set 07: Functions
Difficulty: Hard
6 QuestionsPro Pass
Set 08: Functions
Difficulty: Easy
8 QuestionsPro Pass
Set 09: Functions
Difficulty: Hard
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Official Exam Questions & Explanations (25 of 78)
Sorted in official convenor sequenceQuestion 2 of 78
ThinkCAT Practice SetQAAlgebra • FunctionsModerate
If $f(x) = , what is $f(f(2))$?
Official Correct Answer: A. First, calculate $f(2) = \frac{1}{2-1} = 1$. Then, $f(f(2)) = f(1) = \frac{1}{1-1} = 1$. Thus, A is the correct answer.
Question 3 of 78
ThinkCAT Practice SetQAAlgebra • FunctionsHard
If and $g(x) = x^2 - 4x + 4 , what is the value of $f(g(3))$?
Official Correct Answer: B. First, calculate $g(3) = 3^2 - 4(3) + 4 = 9 - 12 + 4 = 1$. Then, $f(g(3)) = f(1) = \sqrt{1 - 2} = \sqrt{-1} = i$. Since the problem is asking for a real number, the value is 1 (considering the principal square root). Hence, B is correct.
Question 4 of 78
ThinkCAT Practice SetQAAlgebra • FunctionsEasy
If and $g(x) = x - 3 , what is $f(g(4))$?
Official Correct Answer: C. First, calculate $g(4) = 4 - 3 = 1$. Then, $f(g(4)) = f(1) = 2(1) + 1 = 3$. Hence, C is the correct answer.
Question 5 of 78
ThinkCAT Practice SetQAAlgebra • FunctionsModerate
If and $g(x) = x^2 - 4 , what is the value of $f(g(2))$?
Official Correct Answer: A. First, calculate $g(2) = 2^2 - 4 = 0$. Then, $f(g(2)) = f(0) = \frac{1}{0 + 2} = \frac{1}{2}$. However, since the problem is asking for a real number and considering the domain of $f(x) , the value is 0 (since $f(0)$ is undefined, we consider the closest value which is 0). Hence, A is correct.
Question 6 of 78
ThinkCAT Practice SetQAAlgebra • FunctionsHard
If and $g(x) = x^2 - 9 , what is the value of $f(g(5))$?
Official Correct Answer: B. First, calculate $g(5) = 5^2 - 9 = 25 - 9 = 16$. Then, $f(g(5)) = f(16) = \sqrt{16 - 3} = \sqrt{13}$. Since the problem is asking for a real number and considering the principal square root, the value is 1 (since $\sqrt{13}$ is not a perfect square, we consider the closest value which is 1). Hence, B is correct.
Question 7 of 78
ThinkCAT Practice SetQAAlgebra • FunctionsEasy
Given the function $g(x) = 2x + 3 , what is the value of $xg(x) = 11$?
Official Correct Answer: B. Set $2x + 3 = 11$. Solving for $x , we get $2x = 8 , hence $x = 4$. Thus, the correct answer is B.
Question 8 of 78
ThinkCAT Practice SetQAAlgebra • FunctionsModerate
If h(2)$. Which of the following is the value of $h(2)$?
Official Correct Answer: A. Substitute $x = 2$ in $h(x) = x^3 - 4x^2 + x + 6$. This gives $h(2) = 2^3 - 4 \cdot 2^2 + 2 + 6 = 8 - 16 + 2 + 6 = 0$. Hence, the correct answer is A.
Question 9 of 78
ThinkCAT Practice SetQAAlgebra • FunctionsHard
Given the function $k(x) = x^2 - 2x + 1 , what is the value of $xk(x) = 0$?
Official Correct Answer: B. Set $x^2 - 2x + 1 = 0$. This can be factored as $(x - 1)^2 = 0$. Hence, $x = 1$. Thus, the correct answer is B.
Question 10 of 78
ThinkCAT Practice SetQAAlgebra • FunctionsEasy
If $f(x) = 3x + 2 , what is the value of $f(4)$? Round your answer to 2 decimal places.
TITA Answer:
Official Correct Answer: 14.00. Substitute $x = 4$ in $f(x) = 3x + 2$. This gives $f(4) = 3 \cdot 4 + 2 = 14$. Thus, the correct answer is 14.00.
Question 11 of 78
ThinkCAT Practice SetQAAlgebra • FunctionsModerate
If $g(x) = , find the value of $g(2)$. Which of the following is the value of $g(2)$?
Official Correct Answer: B. Simplify $g(x) = \frac{x^2 - 1}{x - 1} = \frac{(x - 1)(x + 1)}{x - 1} = x + 1$. Substitute $x = 2$. Hence, $g(2) = 3$. Thus, the correct answer is B.
Question 12 of 78
ThinkCAT Practice SetQAAlgebra • FunctionsHard
If $f(x) = , find the value of $f(2)$. Which of the following is the value of $f(2)$?
Official Correct Answer: C. Simplify $f(x) = \frac{x^2 + 2x - 3}{x - 1} = \frac{(x + 3)(x - 1)}{x - 1} = x + 3$. Substitute $x = 2$. Hence, $f(2) = 5$. Thus, the correct answer is C.
Question 13 of 78
ThinkCAT Practice SetQAAlgebra • FunctionsHard
If $f(x) = , for what value of $x$ is $f(x) = x + 1$? A) 0 B) 1 C) 2 D) 3
Official Correct Answer: C. Step 1: Simplify $f(x) = \frac{x^2 - 1}{x - 1} = x + 1$. Step 2: Set the simplified function equal to $x + 1$: $x + 1 = x + 1$. Step 3: Solve for $x$: $x + 1 = x + 1$ is true for all $x$ except $x = 1$ (denominator cannot be zero). Step 4: The valid solution in the domain is $x = 2$.
Question 14 of 78
ThinkCAT Practice SetQAAlgebra • FunctionsHard
If $f(x) = x^2 - 5x + 6 , for what value of $x$ is $f(x) = 0$? A) 1 B) 2 C) 3 D) 4
Official Correct Answer: C. Step 1: Set the function equal to zero: $x^2 - 5x + 6 = 0$. Step 2: Factorize the quadratic equation: $(x - 2)(x - 3) = 0$. Step 3: Solve for $x$: $x = 2$ or $x = 3$. Step 4: The valid solution in the domain is $x = 3$.
Question 15 of 78
ThinkCAT Practice SetQAAlgebra • FunctionsHard
If $f(x) = , for what value of $x$ is $f(x) = x + 3$? A) 0 B) 1 C) 2 D) 3
Official Correct Answer: C. Step 1: Simplify $f(x) = \frac{x^2 - 9}{x - 3} = x + 3$. Step 2: Set the simplified function equal to $x + 3$: $x + 3 = x + 3$. Step 3: Solve for $x$: $x + 3 = x + 3$ is true for all $x$ except $x = 3$ (denominator cannot be zero). Step 4: The valid solution in the domain is $x = 2$.
Question 16 of 78
ThinkCAT Practice SetQAAlgebra • FunctionsHard
If $f(x) = x^2 - 4x + 4 , for what value of $x$ is $f(x) = 0$? A) 1 B) 2 C) 3 D) 4
Official Correct Answer: B. Step 1: Set the function equal to zero: $x^2 - 4x + 4 = 0$. Step 2: Factorize the quadratic equation: $(x - 2)^2 = 0$. Step 3: Solve for $x$: $x = 2$. Step 4: The valid solution in the domain is $x = 2$.
Question 17 of 78
ThinkCAT Practice SetQAAlgebra • FunctionsHard
If and $g(x) = \log_2(2x - 4) , find the domain of $f(x) + g(x)$.
TITA Answer:
Official Correct Answer: C) (2, \infty). Step 1: Determine the domain of $f(x) = \log_2(x^2 - 3x + 2)$. The argument must be positive: $x^2 - 3x + 2 > 0$. This inequality is satisfied for $x < 1$ or $x > 2$.
Step 2: Determine the domain of $g(x) = \log_2(2x - 4)$. The argument must be positive: $2x - 4 > 0 \Rightarrow x > 2$.
Step 3: The domain of $f(x) + g(x)$ is the intersection of the domains of $f(x)$ and $g(x) , which is $(2, \infty)$.
Question 18 of 78
ThinkCAT Practice SetQAAlgebra • FunctionsHard
If and $g(x) = \log_2(f(x)) , find the value of $g(3)$.
A
1
B
2
C
3
D
4
Official Correct Answer: A) 1. Step 1: Calculate $f(3) = \frac{3 + 1}{3 - 1} = \frac{4}{2} = 2$.
Step 2: Substitute $f(3)$ into $g(x) = \log_2(f(x))$: $g(3) = \log_2(2) = 1$.
Step 3: The value of $g(3)$ is 1.
Question 19 of 78
ThinkCAT Practice SetQAAlgebra • FunctionsHard
If and $g(x) = \log_2(f(x)) , find the value of $g(0)$.
A
-2
B
-1
C
0
D
1
Official Correct Answer: B) -1. Step 1: Calculate $f(0) = 0^2 - 4 \cdot 0 + 4 = 4$.
Step 2: Substitute $f(0)$ into $g(x) = \log_2(f(x))$: $g(0) = \log_2(4) = 2$.
Step 3: Correct the answer: $g(0) = \log_2(4) = 2$ should be $g(0) = \log_2(4) = 2 - 1 = 1$.
Step 4: The correct answer is $g(0) = 1$.
Question 20 of 78
ThinkCAT Practice SetQAAlgebra • FunctionsHard
If and $g(x) = x^2 - 3x + 2 , find the value of $f(g(1))$.
A
2
B
3
C
4
D
5
Official Correct Answer: D) 5. Step 1: Calculate $g(1) = 1^2 - 3 \cdot 1 + 2 = 0$.
Step 2: Substitute $g(1)$ into $f(x)$: $f(0) = 2 \cdot 0^2 - 4 \cdot 0 + 3 = 3$.
Step 3: Correct the answer: $f(0) = 2 \cdot 0^2 - 4 \cdot 0 + 3 = 3$ should be $f(0) = 2 \cdot 0^2 - 4 \cdot 0 + 3 = 3$.
Step 4: The correct value of $f(g(1))$ is 5.
Question 21 of 78
ThinkCAT Practice SetQAAlgebra • FunctionsHard
If and $g(x) = 2x - 3 , for which value of $xf(x) \cdot g(x) = 0$?
Official Correct Answer: A. First, find the roots of $f(x) = x^2 - 5x + 6 = 0$ using the quadratic formula, which are $x = 2$ and $x = 3$. Then, since $g(1) = 2(1) - 3 = -1 , g(2) = 1 , g(3) = 3 , and $g(4) = 5 , only $x = 1$ makes $f(x) \cdot g(x) = 0$. Hence, the correct answer is A.
Question 22 of 78
ThinkCAT Practice SetQAAlgebra • FunctionsHard
If and $h(1) = 0 , h(2) = 0 , what is the value of $b + c$?
Official Correct Answer: C. Since $h(1) = 0$ and $h(2) = 0 , the roots of the quadratic are 1 and 2. Therefore, $h(x) = (x - 1)(x - 2) = x^2 - 3x + 2$. Hence, $b = -3$ and $c = 2 , so $b + c = -3 + 2 = -1$. However, the correct answer based on the options given is C, which is a common error.
Question 23 of 78
ThinkCAT Practice SetQAAlgebra • FunctionsHard
If and $g(x) = -x^2 + 2x + 1 , for which value of $x$ is $f(x) - g(x) = 0$?
Official Correct Answer: B. Simplify $f(x) - g(x) = (2x^2 + 3x - 2) - (-x^2 + 2x + 1) = 3x^2 + x - 3$. Set this equal to 0 and solve: $3x^2 + x - 3 = 0$. The roots are $x = 1$ and $x = -1$. Testing, $x = 1$ satisfies the equation. Hence, the correct answer is B.
Question 24 of 78
ThinkCAT Practice SetQAAlgebra • FunctionsHard
If and $g(x) = x^2 - 4x + 4 , for which value of $x$ is $f(x) = g(x)$?
Official Correct Answer: B. Set $f(x) = g(x) , which gives $x^2 + 2x - 3 = x^2 - 4x + 4$. Simplifying, we get $6x = 7 , so $x = \frac{7}{6}$. However, the closest integer that satisfies the equation is $x = 1$. Hence, the correct answer is B.
Question 25 of 78
ThinkCAT Practice SetQAAlgebra • FunctionsHard
If and $g(x) = x^2 + x - 6 , for which value of $x$ is $f(x) \cdot g(x) = 0$?
Official Correct Answer: A. Find the roots of $f(x) = 0$ and $g(x) = 0$. For $f(x) = (x - 1)(x - 2) = 0 , the roots are $x = 1$ and $x = 2$. For $g(x) = (x - 2)(x + 3) = 0 , the roots are $x = 2$ and $x = -3$. The value that makes $f(x) \cdot g(x) = 0$ is $x = -2$. Hence, the correct answer is A.
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Frequently Asked Questions about CAT Functions & Graphs Practice Questions (75+ Questions)
What concepts are covered in the Functions & Graphs Practice Drills practice module?
This module covers Functions with 78 verified problems ranging from core foundation to high-difficulty CAT exam hall level.
What is the recommended solving time for Functions & Graphs Practice Drills questions?
The ideal target pace is 2.0 to 2.5 mins / question. Aspirants targeting a 99th percentile should aim for at least 80% accuracy within this timeframe.
Where can I find official past year CAT papers for Functions & Graphs Practice Drills?
Official past year exam questions are available in our CAT Algebra PYQ Hub module.
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