Answer :
- First, find the value of $g(2)$ by substituting $x=2$ into $g(x)$: $g(2) = 2^3 + 1 = 9$.
- Next, substitute the value of $g(2)$ into $f(x)$: $f(g(2)) = f(9) = 3(9)^2 - 7$.
- Simplify the expression: $3(81) - 7 = 243 - 7 = 236$.
- The final answer is $\boxed{236}$.
### Explanation
1. Understanding the Problem
We are given two functions: $f(x) = 3x^2 - 7$ and $g(x) = x^3 + 1$. Our goal is to find the value of the composite function $f(g(2))$. This means we first need to evaluate $g(2)$, and then substitute that result into $f(x)$.
2. Evaluating g(2)
First, let's find $g(2)$. We substitute $x = 2$ into the expression for $g(x)$: $$g(2) = (2)^3 + 1 = 8 + 1 = 9.$$
3. Evaluating f(g(2))
Now that we have $g(2) = 9$, we can find $f(g(2))$ by substituting $9$ into the expression for $f(x)$: $$f(g(2)) = f(9) = 3(9)^2 - 7 = 3(81) - 7 = 243 - 7 = 236.$$
4. Final Answer
Therefore, $f(g(2)) = 236$.
### Examples
Composite functions are useful in many real-world scenarios. For example, consider a store that offers a discount on all items and then applies a sales tax. If $g(x)$ represents the price after the discount and $f(x)$ represents the price after the sales tax, then $f(g(x))$ represents the final price you pay. Understanding composite functions helps in analyzing such multi-step processes.
- Next, substitute the value of $g(2)$ into $f(x)$: $f(g(2)) = f(9) = 3(9)^2 - 7$.
- Simplify the expression: $3(81) - 7 = 243 - 7 = 236$.
- The final answer is $\boxed{236}$.
### Explanation
1. Understanding the Problem
We are given two functions: $f(x) = 3x^2 - 7$ and $g(x) = x^3 + 1$. Our goal is to find the value of the composite function $f(g(2))$. This means we first need to evaluate $g(2)$, and then substitute that result into $f(x)$.
2. Evaluating g(2)
First, let's find $g(2)$. We substitute $x = 2$ into the expression for $g(x)$: $$g(2) = (2)^3 + 1 = 8 + 1 = 9.$$
3. Evaluating f(g(2))
Now that we have $g(2) = 9$, we can find $f(g(2))$ by substituting $9$ into the expression for $f(x)$: $$f(g(2)) = f(9) = 3(9)^2 - 7 = 3(81) - 7 = 243 - 7 = 236.$$
4. Final Answer
Therefore, $f(g(2)) = 236$.
### Examples
Composite functions are useful in many real-world scenarios. For example, consider a store that offers a discount on all items and then applies a sales tax. If $g(x)$ represents the price after the discount and $f(x)$ represents the price after the sales tax, then $f(g(x))$ represents the final price you pay. Understanding composite functions helps in analyzing such multi-step processes.
