If `f(x)=x^n , n in Nandgof(x)=ng(x)` then g(x) can be

Options

A. `n|x|`
B. `3x^(1//3)`
C. `e^(x)`
D. `log|x|`

Answer

To solve the problem step by step, we need to analyze the given functions and the condition provided.

### Step 1: Understand the given functions
We have:
- \( f(x) = x^n \) where \( n \in \mathbb{N} \) (natural numbers).
- The condition \( g(f(x)) = n \cdot g(x) \).

### Step 2: Substitute \( f(x) \) into the condition
Substituting \( f(x) \) into the condition gives us:
\[ g(x^n) = n \cdot g(x) \]

### Step 3: Analyze the options for \( g(x) \)
We need to check which of the given options satisfies the equation \( g(x^n) = n \cdot g(x) \).

#### Option 1: \( g(x) = n \cdot |x| \)
Substituting into the equation:
\[ g(x^n) = n \cdot |x^n| = n \cdot |x|^n \]
And,
\[ n \cdot g(x) = n \cdot (n \cdot |x|) = n^2 \cdot |x| \]
These are not equal for all \( x \), so this option is not valid.

#### Option 2: \( g(x) = 3x^{1/3} \)
Substituting into the equation:
\[ g(x^n) = 3(x^n)^{1/3} = 3x^{n/3} \]
And,
\[ n \cdot g(x) = n \cdot (3x^{1/3}) = 3n \cdot x^{1/3} \]
These are not equal for all \( x \), so this option is not valid.

#### Option 3: \( g(x) = e^x \)
Substituting into the equation:
\[ g(x^n) = e^{x^n} \]
And,
\[ n \cdot g(x) = n \cdot e^x \]
These are not equal for all \( x \), so this option is not valid.

#### Option 4: \( g(x) = \log |x| \)
Substituting into the equation:
\[ g(x^n) = \log |x^n| = n \cdot \log |x| \]
And,
\[ n \cdot g(x) = n \cdot \log |x| \]
These are equal, so this option is valid.

### Conclusion
The function \( g(x) \) that satisfies the condition \( g(f(x)) = n \cdot g(x) \) is:
\[ g(x) = \log |x| \]

### Final Answer:
The correct option is \( g(x) = \log |x| \).

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