If `f(x)=x^(3)-x and phi (x)= sin 2x `, then

Options

A. `phi(f(2))=sin2`
B. `phi(f(1))=1`
C. `f(phi(pi//12))=-(3)/(8)`
D. `f(f(1))=2`

Answer

To solve the problem, we need to evaluate the functions \( f(x) = x^3 - x \) and \( \phi(x) = \sin(2x) \) for specific values and combinations. Let's go through the steps systematically.

### Step 1: Evaluate \( f(\phi(x)) \)

We start by substituting \( \phi(x) \) into \( f(x) \):

\[ f(\phi(x)) = f(\sin(2x)) = (\sin(2x))^3 - \sin(2x) \]

### Step 2: Simplify \( f(\phi(x)) \)

Now we simplify the expression we obtained:

\[ f(\phi(x)) = \sin^3(2x) - \sin(2x) \]

### Step 3: Evaluate \( f(\phi(2)) \)

Next, we need to evaluate \( f(\phi(2)) \):

1. Calculate \( \phi(2) \):
\[ \phi(2) = \sin(2 \cdot 2) = \sin(4) \]

2. Now substitute \( \phi(2) \) into \( f \):
\[ f(\phi(2)) = f(\sin(4)) = (\sin(4))^3 - \sin(4) \]

### Step 4: Evaluate \( f(\phi(1)) \)

Next, we evaluate \( f(\phi(1)) \):

1. Calculate \( \phi(1) \):
\[ \phi(1) = \sin(2 \cdot 1) = \sin(2) \]

2. Substitute \( \phi(1) \) into \( f \):
\[ f(\phi(1)) = f(\sin(2)) = (\sin(2))^3 - \sin(2) \]

### Step 5: Evaluate \( f(\phi(\frac{\pi}{12})) \)

Now we evaluate \( f(\phi(\frac{\pi}{12})) \):

1. Calculate \( \phi(\frac{\pi}{12}) \):
\[ \phi\left(\frac{\pi}{12}\right) = \sin\left(2 \cdot \frac{\pi}{12}\right) = \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \]

2. Substitute \( \phi\left(\frac{\pi}{12}\right) \) into \( f \):
\[ f\left(\phi\left(\frac{\pi}{12}\right)\right) = f\left(\frac{1}{2}\right) = \left(\frac{1}{2}\right)^3 - \frac{1}{2} = \frac{1}{8} - \frac{1}{2} = \frac{1}{8} - \frac{4}{8} = -\frac{3}{8} \]

### Step 6: Evaluate \( f(f(1)) \)

Finally, we evaluate \( f(f(1)) \):

1. Calculate \( f(1) \):
\[ f(1) = 1^3 - 1 = 0 \]

2. Substitute \( f(1) \) into \( f \):
\[ f(f(1)) = f(0) = 0^3 - 0 = 0 \]

### Summary of Results

- \( f(\phi(2)) = (\sin(4))^3 - \sin(4) \)
- \( f(\phi(1)) = (\sin(2))^3 - \sin(2) \)
- \( f(\phi(\frac{\pi}{12})) = -\frac{3}{8} \)
- \( f(f(1)) = 0 \)