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Identify the type of the functions: f(x)...

Identify the type of the functions: `f(x)={g(x)-g(-x)}^(3)`

A

Odd

B

Even

C

Neither

D

Both

Text Solution

AI Generated Solution

The correct Answer is:
To identify the type of the function \( f(x) = (g(x) - g(-x))^3 \), we will follow these steps: ### Step 1: Calculate \( f(-x) \) We start by substituting \(-x\) into the function: \[ f(-x) = (g(-x) - g(-(-x)))^3 \] This simplifies to: \[ f(-x) = (g(-x) - g(x))^3 \] ### Step 2: Rearranging the expression Notice that \( g(-x) - g(x) \) can be rewritten as: \[ f(-x) = -(g(x) - g(-x))^3 \] This is because \( g(-x) - g(x) = - (g(x) - g(-x)) \). ### Step 3: Relate \( f(-x) \) to \( f(x) \) Now we can express \( f(-x) \) in terms of \( f(x) \): \[ f(-x) = - (g(x) - g(-x))^3 = -f(x) \] ### Step 4: Conclusion about the function type From the relationship \( f(-x) = -f(x) \), we conclude that the function \( f(x) \) is an **odd function**. ### Summary Thus, the function \( f(x) = (g(x) - g(-x))^3 \) is an odd function. ---
To identify the type of the function \( f(x) = (g(x) - g(-x))^3 \), we will follow these steps: ### Step 1: Calculate \( f(-x) \) We start by substituting \(-x\) into the function: \[ f(-x) = (g(-x) - g(-(-x)))^3 \] ...
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Knowledge Check

  • If f(x)=cosx and g(x)=2x+1 , which of the following are even functions? I. f(x)*g(x) II f(g(x)) III. g(f(x))

    A
    only I
    B
    only II
    C
    only III
    D
    Only I and II

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