Waste less time on Facebook — follow Brilliant.
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Back to all chapters
Functional Equations
Functional equations are equations where the unknowns are the functions. Rather than solving for x, you solve for the function in questions like "Find all functions that have these properties."
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Your answer seems reasonable.
Find out if you're right!
Sign up to access problem solutions.
That seems reasonable.
Find out if you're right!
Concept Quizzes
Define the function \( f^1(x) = \frac{1}{1-x} \), and \( f^n (x) = f^1 ( f^{n-1} (x) ) \) for positive integers \(n \).
Evaluate \( f^{36} ( 10 ) . \)
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Your answer seems reasonable.
Find out if you're right!
Sign up to access problem solutions.
That seems reasonable.
Find out if you're right!
by Brilliant Staff
Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that
\( 5 f( x + y) + y^ 5 = f(x) + (x+y) ^ 5. \)
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Your answer seems reasonable.
Find out if you're right!
Sign up to access problem solutions.
That seems reasonable.
Find out if you're right!
by Brilliant Staff
Suppose that function \(f\) satisfies
\[f\left(\frac{x+y}{2}\right)=\frac{f(x)+f(y)}{2} \text{ and } f(0)=17.\]
Which of the following represents the family of solutions for \( f(x) \)?
\(k\) in the choices below is a constant.
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Your answer seems reasonable.
Find out if you're right!
Sign up to access problem solutions.
That seems reasonable.
Find out if you're right!
by Brilliant Staff
\(P(x) \) is a monic polynomial of degree \( 2017 \) If \[P(0)=2016, P(1)=2015, P(2)=2014, \ldots, P(2016)=0,\] what is the value of \(P(2017)?\)
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Your answer seems reasonable.
Find out if you're right!
Sign up to access problem solutions.
That seems reasonable.
Find out if you're right!
by Brilliant Staff
If function \(f\) satisfies \(f(2)=12\) and \[f(x+y)=f(x)+f(y)\] for all real numbers \(x\) and \(y,\) what is the value of \(f(-3)?\)
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Your answer seems reasonable.
Find out if you're right!
Sign up to access problem solutions.
That seems reasonable.
Find out if you're right!
by Brilliant Staff