When ` f: R to R and g: R to R ` are two functions defined by `f(x) = 8x^(3) and g(x) = = x^((1)/(3))` respectively and (g of ) (x) = k (f o g ) (x) , then k is

Options

A. `4`
B. ` 2`
C. ` (1)/(2)`
D. None of these

Answer

To solve the problem, we need to find the value of \( k \) in the equation \( g(f(x)) = k(f(g(x))) \) given the functions \( f(x) = 8x^3 \) and \( g(x) = x^{1/3} \).

### Step 1: Calculate \( g(f(x)) \)

We start by substituting \( f(x) \) into \( g(x) \):
\[ g(f(x)) = g(8x^3) \]
Since \( g(x) = x^{1/3} \), we have:
\[ g(8x^3) = (8x^3)^{1/3} \]
Using the property of exponents, we simplify:
\[ (8x^3)^{1/3} = 8^{1/3} \cdot (x^3)^{1/3} = 2 \cdot x = 2x \]

### Step 2: Calculate \( f(g(x)) \)

Next, we substitute \( g(x) \) into \( f(x) \):
\[ f(g(x)) = f(x^{1/3}) \]
Using the definition of \( f(x) \):
\[ f(x^{1/3}) = 8(x^{1/3})^3 \]
Again, simplifying gives:
\[ 8(x^{1/3})^3 = 8x \]

### Step 3: Set up the equation

Now we have:
\[ g(f(x)) = 2x \quad \text{and} \quad f(g(x)) = 8x \]
According to the problem statement:
\[ g(f(x)) = k(f(g(x))) \]
Substituting the expressions we found:
\[ 2x = k(8x) \]

### Step 4: Solve for \( k \)

To isolate \( k \), we divide both sides by \( 8x \) (assuming \( x \neq 0 \)):
\[ k = \frac{2x}{8x} = \frac{2}{8} = \frac{1}{4} \]

Thus, the value of \( k \) is:
\[ \boxed{\frac{1}{4}} \]