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Let f (x) = |x – 2| + |x – 3| + |x – 4| ...

Let `f (x) = |x – 2| + |x – 3| + |x – 4| and g(x) = f (x + 1)`. Then

A

g(x) is an even function

B

g(x) is an odd function

C

g(x) is neither even nor odd

D

None of these

Text Solution

AI Generated Solution

The correct Answer is:
To determine whether the function \( g(x) = f(x + 1) \) is even, odd, or neither, we first need to analyze the function \( f(x) \). ### Step 1: Define the function \( f(x) \) The function is given as: \[ f(x) = |x - 2| + |x - 3| + |x - 4| \] ### Step 2: Define the function \( g(x) \) Now, we substitute \( x + 1 \) into \( f(x) \) to find \( g(x) \): \[ g(x) = f(x + 1) = |(x + 1) - 2| + |(x + 1) - 3| + |(x + 1) - 4| \] This simplifies to: \[ g(x) = |x - 1| + |x - 2| + |x - 3| \] ### Step 3: Check if \( g(x) \) is even or odd To check if \( g(x) \) is even, we need to evaluate \( g(-x) \): \[ g(-x) = |(-x) - 1| + |(-x) - 2| + |(-x) - 3| = |-x - 1| + |-x - 2| + |-x - 3| \] This can be rewritten as: \[ g(-x) = |-(x + 1)| + |-(x + 2)| + |-(x + 3)| = |x + 1| + |x + 2| + |x + 3| \] ### Step 4: Compare \( g(-x) \) and \( g(x) \) Now we compare \( g(-x) \) with \( g(x) \): - \( g(x) = |x - 1| + |x - 2| + |x - 3| \) - \( g(-x) = |x + 1| + |x + 2| + |x + 3| \) Since \( g(-x) \) is not equal to \( g(x) \) and also not equal to \(-g(x)\), we conclude that \( g(x) \) is neither even nor odd. ### Conclusion Thus, the final answer is that \( g(x) \) is neither even nor odd. ---
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