Answer :
- First, find the composite function $f(f(x))$ by substituting $f(x)$ into $f(x)$.
- Then, expand the expression $(x^3 - 2)^3 - 2$.
- Simplify the expanded expression to get $x^9 - 6x^6 + 12x^3 - 10$.
- The final answer is $\boxed{x^9 - 6x^6 + 12x^3 - 10}$.
### Explanation
1. Understanding the problem
We are given the function $f(x) = x^3 - 2$. We need to find $(f \circ f)(x)$, which means $f(f(x))$.
2. Finding the composite function
To find $f(f(x))$, we substitute $f(x)$ into the expression for $f(x)$. So, we have $f(f(x)) = f(x^3 - 2) = (x^3 - 2)^3 - 2$.
3. Expanding the expression
Now, we need to expand the expression $(x^3 - 2)^3 - 2$. We can use the binomial theorem or direct multiplication. Let's use direct multiplication:
$(x^3 - 2)^3 = (x^3 - 2)(x^3 - 2)(x^3 - 2) = (x^6 - 4x^3 + 4)(x^3 - 2) = x^9 - 2x^6 - 4x^6 + 8x^3 + 4x^3 - 8 = x^9 - 6x^6 + 12x^3 - 8$.
4. Simplifying the expression
Now, we subtract 2 from the expanded expression:
$f(f(x)) = (x^9 - 6x^6 + 12x^3 - 8) - 2 = x^9 - 6x^6 + 12x^3 - 10$.
5. Final Answer
Comparing our result with the given options, we see that it matches option B. Therefore, $(f \circ f)(x) = x^9 - 6x^6 + 12x^3 - 10$.
### Examples
Composite functions are used in various real-world applications, such as in computer graphics to perform a series of transformations on an object (e.g., scaling, rotating, and translating). Each transformation can be represented as a function, and applying multiple transformations in sequence is equivalent to composing those functions. Another example is in economics, where the production cost is a function of the number of items produced, and the number of items sold is a function of the price. The total profit can then be expressed as a composite function of these relationships.
- Then, expand the expression $(x^3 - 2)^3 - 2$.
- Simplify the expanded expression to get $x^9 - 6x^6 + 12x^3 - 10$.
- The final answer is $\boxed{x^9 - 6x^6 + 12x^3 - 10}$.
### Explanation
1. Understanding the problem
We are given the function $f(x) = x^3 - 2$. We need to find $(f \circ f)(x)$, which means $f(f(x))$.
2. Finding the composite function
To find $f(f(x))$, we substitute $f(x)$ into the expression for $f(x)$. So, we have $f(f(x)) = f(x^3 - 2) = (x^3 - 2)^3 - 2$.
3. Expanding the expression
Now, we need to expand the expression $(x^3 - 2)^3 - 2$. We can use the binomial theorem or direct multiplication. Let's use direct multiplication:
$(x^3 - 2)^3 = (x^3 - 2)(x^3 - 2)(x^3 - 2) = (x^6 - 4x^3 + 4)(x^3 - 2) = x^9 - 2x^6 - 4x^6 + 8x^3 + 4x^3 - 8 = x^9 - 6x^6 + 12x^3 - 8$.
4. Simplifying the expression
Now, we subtract 2 from the expanded expression:
$f(f(x)) = (x^9 - 6x^6 + 12x^3 - 8) - 2 = x^9 - 6x^6 + 12x^3 - 10$.
5. Final Answer
Comparing our result with the given options, we see that it matches option B. Therefore, $(f \circ f)(x) = x^9 - 6x^6 + 12x^3 - 10$.
### Examples
Composite functions are used in various real-world applications, such as in computer graphics to perform a series of transformations on an object (e.g., scaling, rotating, and translating). Each transformation can be represented as a function, and applying multiple transformations in sequence is equivalent to composing those functions. Another example is in economics, where the production cost is a function of the number of items produced, and the number of items sold is a function of the price. The total profit can then be expressed as a composite function of these relationships.
