CAT Exam > CAT Questions > f(x) and g(x) are 2 polynomial functions such...
Start Learning for Free
f(x) and g(x) are 2 polynomial functions such that f(g(x)) = x2 + 12x + 33 and g(f(x)) = x2 + 4x + 5. it is given that f(26) = 46. What is maximum possible value of f(3)?
Correct answer is '1'. Can you explain this answer?
Most Upvoted Answer
f(x) and g(x) are 2 polynomial functions such that f(g(x)) = x2 + 12x ...
Given:
- f(x) and g(x) are two polynomial functions.
- f(g(x)) = x^2 + 12x + 33
- g(f(x)) = x^2 - 4x + 5
- f(26) = 46
To find:
The maximum possible value of f(3)
Explanation:
We are given two composite functions f(g(x)) and g(f(x)) and their respective expressions. We need to find the maximum possible value of f(3), given that f(26) = 46.
Step 1: Finding g(x)
To find g(x), we can equate g(f(x)) to the given expression x^2 - 4x + 5:
g(f(x)) = x^2 - 4x + 5
Comparing this with the given expression g(f(x)) = x^2 - 4x + 5, we can conclude that f(x) = x^2 - 4x + 5.
Step 2: Finding g(x)
Now that we have the expression for f(x), we can substitute it into f(g(x)):
f(g(x)) = x^2 + 12x + 33
Substituting g(x) = x^2 - 4x + 5 into f(g(x)), we get:
f(x^2 - 4x + 5) = x^2 + 12x + 33
Let's simplify this equation to find f(x).
Step 3: Simplifying the equation
Expanding the function f(x^2 - 4x + 5), we get:
f(x^2 - 4x + 5) = (x^2 - 4x + 5)^2 - 4(x^2 - 4x + 5) + 5
Expanding further:
f(x^2 - 4x + 5) = x^4 - 8x^3 + 29x^2 - 76x + 45
Comparing this expression with f(g(x)) = x^2 + 12x + 33, we can conclude that:
f(x) = x^4 - 8x^3 + 29x^2 - 76x + 45
Step 4: Finding f(3)
To find f(3), we substitute x = 3 into the expression for f(x):
f(3) = 3^4 - 8(3)^3 + 29(3)^2 - 76(3) + 45
= 81 - 216 + 261 - 228 + 45
= -57
Therefore, the maximum possible value of f(3) is -57.
- f(x) and g(x) are two polynomial functions.
- f(g(x)) = x^2 + 12x + 33
- g(f(x)) = x^2 - 4x + 5
- f(26) = 46
To find:
The maximum possible value of f(3)
Explanation:
We are given two composite functions f(g(x)) and g(f(x)) and their respective expressions. We need to find the maximum possible value of f(3), given that f(26) = 46.
Step 1: Finding g(x)
To find g(x), we can equate g(f(x)) to the given expression x^2 - 4x + 5:
g(f(x)) = x^2 - 4x + 5
Comparing this with the given expression g(f(x)) = x^2 - 4x + 5, we can conclude that f(x) = x^2 - 4x + 5.
Step 2: Finding g(x)
Now that we have the expression for f(x), we can substitute it into f(g(x)):
f(g(x)) = x^2 + 12x + 33
Substituting g(x) = x^2 - 4x + 5 into f(g(x)), we get:
f(x^2 - 4x + 5) = x^2 + 12x + 33
Let's simplify this equation to find f(x).
Step 3: Simplifying the equation
Expanding the function f(x^2 - 4x + 5), we get:
f(x^2 - 4x + 5) = (x^2 - 4x + 5)^2 - 4(x^2 - 4x + 5) + 5
Expanding further:
f(x^2 - 4x + 5) = x^4 - 8x^3 + 29x^2 - 76x + 45
Comparing this expression with f(g(x)) = x^2 + 12x + 33, we can conclude that:
f(x) = x^4 - 8x^3 + 29x^2 - 76x + 45
Step 4: Finding f(3)
To find f(3), we substitute x = 3 into the expression for f(x):
f(3) = 3^4 - 8(3)^3 + 29(3)^2 - 76(3) + 45
= 81 - 216 + 261 - 228 + 45
= -57
Therefore, the maximum possible value of f(3) is -57.
Community Answer
f(x) and g(x) are 2 polynomial functions such that f(g(x)) = x2 + 12x ...
We know that f(g(x)) =x2 + 12x +33
Thus f(g(f(x))) = f(x2) + 12f(x) + 33
Putting g(f(x)) = x2 + 4x + 5
Thus f(x2 + 4x + 5) = f(x)2 + 12f(x) + 33
Given f(26) = 46
putting x2 + 4x + 5 = 26
Or x2 + 4x - 21 = 0
Or (x+7)(x-3) =0.
Putting x = 3 in f(x2 + 4x + 5) = f(x)2 + 12f(x) + 33
f(26) = 46 = f(3)2+12f(3)+33
Or f(3)2 + 12f(3) - 13 = 0
(f(3)+13)(f(3)-1) = 0 Thus f(3) = 1 or -13
1 > -13 so it is the answer
Thus f(g(f(x))) = f(x2) + 12f(x) + 33
Putting g(f(x)) = x2 + 4x + 5
Thus f(x2 + 4x + 5) = f(x)2 + 12f(x) + 33
Given f(26) = 46
putting x2 + 4x + 5 = 26
Or x2 + 4x - 21 = 0
Or (x+7)(x-3) =0.
Putting x = 3 in f(x2 + 4x + 5) = f(x)2 + 12f(x) + 33
f(26) = 46 = f(3)2+12f(3)+33
Or f(3)2 + 12f(3) - 13 = 0
(f(3)+13)(f(3)-1) = 0 Thus f(3) = 1 or -13
1 > -13 so it is the answer
|
Explore Courses for CAT exam
|
|
Similar CAT Doubts
Top Courses for CATView all
f(x) and g(x) are 2 polynomial functions such that f(g(x)) = x2 + 12x + 33and g(f(x)) = x2 + 4x + 5.it is given that f(26) = 46. What is maximum possible value of f(3)?Correct answer is '1'. Can you explain this answer?
Question Description
f(x) and g(x) are 2 polynomial functions such that f(g(x)) = x2 + 12x + 33and g(f(x)) = x2 + 4x + 5.it is given that f(26) = 46. What is maximum possible value of f(3)?Correct answer is '1'. Can you explain this answer? for CAT 2025 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about f(x) and g(x) are 2 polynomial functions such that f(g(x)) = x2 + 12x + 33and g(f(x)) = x2 + 4x + 5.it is given that f(26) = 46. What is maximum possible value of f(3)?Correct answer is '1'. Can you explain this answer? covers all topics & solutions for CAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for f(x) and g(x) are 2 polynomial functions such that f(g(x)) = x2 + 12x + 33and g(f(x)) = x2 + 4x + 5.it is given that f(26) = 46. What is maximum possible value of f(3)?Correct answer is '1'. Can you explain this answer?.
f(x) and g(x) are 2 polynomial functions such that f(g(x)) = x2 + 12x + 33and g(f(x)) = x2 + 4x + 5.it is given that f(26) = 46. What is maximum possible value of f(3)?Correct answer is '1'. Can you explain this answer? for CAT 2025 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about f(x) and g(x) are 2 polynomial functions such that f(g(x)) = x2 + 12x + 33and g(f(x)) = x2 + 4x + 5.it is given that f(26) = 46. What is maximum possible value of f(3)?Correct answer is '1'. Can you explain this answer? covers all topics & solutions for CAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for f(x) and g(x) are 2 polynomial functions such that f(g(x)) = x2 + 12x + 33and g(f(x)) = x2 + 4x + 5.it is given that f(26) = 46. What is maximum possible value of f(3)?Correct answer is '1'. Can you explain this answer?.
Solutions for f(x) and g(x) are 2 polynomial functions such that f(g(x)) = x2 + 12x + 33and g(f(x)) = x2 + 4x + 5.it is given that f(26) = 46. What is maximum possible value of f(3)?Correct answer is '1'. Can you explain this answer? in English & in Hindi are available as part of our courses for CAT.
Download more important topics, notes, lectures and mock test series for CAT Exam by signing up for free.
Here you can find the meaning of f(x) and g(x) are 2 polynomial functions such that f(g(x)) = x2 + 12x + 33and g(f(x)) = x2 + 4x + 5.it is given that f(26) = 46. What is maximum possible value of f(3)?Correct answer is '1'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
f(x) and g(x) are 2 polynomial functions such that f(g(x)) = x2 + 12x + 33and g(f(x)) = x2 + 4x + 5.it is given that f(26) = 46. What is maximum possible value of f(3)?Correct answer is '1'. Can you explain this answer?, a detailed solution for f(x) and g(x) are 2 polynomial functions such that f(g(x)) = x2 + 12x + 33and g(f(x)) = x2 + 4x + 5.it is given that f(26) = 46. What is maximum possible value of f(3)?Correct answer is '1'. Can you explain this answer? has been provided alongside types of f(x) and g(x) are 2 polynomial functions such that f(g(x)) = x2 + 12x + 33and g(f(x)) = x2 + 4x + 5.it is given that f(26) = 46. What is maximum possible value of f(3)?Correct answer is '1'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice f(x) and g(x) are 2 polynomial functions such that f(g(x)) = x2 + 12x + 33and g(f(x)) = x2 + 4x + 5.it is given that f(26) = 46. What is maximum possible value of f(3)?Correct answer is '1'. Can you explain this answer? tests, examples and also practice CAT tests.
|
|
Explore Courses for CAT exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup