State whether the following statement is true or false.
For a function [tex]$f , f ( x + h )= f ( x )+ f ( h )$[/tex].
Choose the correct answer below.
False
True
Answer (2)
Assume the statement is true and try to prove it.
Find a counterexample to disprove the statement.
Consider the function f ( x ) = x 2 .
Calculate f ( x + h ) = ( x + h ) 2 = x 2 + 2 x h + h 2 and f ( x ) + f ( h ) = x 2 + h 2 .
Since f ( x + h ) e q f ( x ) + f ( h ) in general, the statement is false. F a l se
Explanation
Understanding the Problem The statement claims that for any function f , the equation f ( x + h ) = f ( x ) + f ( h ) holds true. We need to determine if this statement is universally true or false.
Finding a Counterexample To show that the statement is false, we only need to find one counterexample, i.e., a function for which the equation does not hold. Let's consider a simple function, such as f ( x ) = x 2 .
Calculating f(x+h) If f ( x ) = x 2 , then f ( x + h ) = ( x + h ) 2 . Expanding this, we get f ( x + h ) = x 2 + 2 x h + h 2 .
Calculating f(x) + f(h) Now let's calculate f ( x ) + f ( h ) . Since f ( x ) = x 2 , we have f ( x ) + f ( h ) = x 2 + h 2 .
Comparing the Results Comparing f ( x + h ) = x 2 + 2 x h + h 2 and f ( x ) + f ( h ) = x 2 + h 2 , we see that they are not equal in general. The difference is the term 2 x h . For example, if x = 1 and h = 1 , then f ( x + h ) = ( 1 + 1 ) 2 = 4 , while f ( x ) + f ( h ) = 1 2 + 1 2 = 2 . Since 4 e q 2 , the statement is false for the function f ( x ) = x 2 .
Conclusion Therefore, the statement "For a function f , f ( x + h ) = f ( x ) + f ( h ) " is false.
Examples
Understanding whether a function satisfies the property f ( x + h ) = f ( x ) + f ( h ) is crucial in many areas of mathematics and physics. For example, linear functions of the form f ( x ) = a x satisfy this property, which is essential in linear algebra and Fourier analysis. However, many other functions, like f ( x ) = x 2 , do not satisfy this property. Recognizing which functions have this property helps in simplifying complex problems and understanding the underlying mathematical structures.
The statement f ( x + h ) = f ( x ) + f ( h ) is false because a simple counterexample, such as the function f ( x ) = x 2 , shows that the two sides of the equation do not generally equal each other. Therefore, the answer is 'False'.
;
