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Answer :
- First, find the value of $g(4)$: $g(4) = 2 \times 4 = 8$.
- Then, substitute $g(4)$ into $f(x)$: $f(g(4)) = f(8) = 3(8)^2 - 3(8) + 6$.
- Calculate $f(8)$: $f(8) = 3(64) - 24 + 6 = 192 - 24 + 6 = 174$.
- Therefore, $f(g(4)) = \boxed{174}$.
### Explanation
1. Understanding the Problem
We are given two functions: $f(x) = 3x^2 - 3x + 6$ and $g(x) = 2x$. Our goal is to find the value of the composite function $f(g(4))$. This means we first need to evaluate $g(4)$, and then substitute that result into the function $f(x)$.
2. Evaluating g(4)
First, let's find $g(4)$. We substitute $x=4$ into the expression for $g(x)$: $$g(4) = 2 \times 4 = 8$$
3. Substituting into f(x)
Now that we have $g(4) = 8$, we can find $f(g(4))$ by substituting $8$ into the function $f(x)$:$$f(g(4)) = f(8) = 3(8)^2 - 3(8) + 6$$
4. Calculating f(8)
Let's calculate $f(8)$:$$f(8) = 3(64) - 24 + 6 = 192 - 24 + 6 = 168 + 6 = 174$$Therefore, $f(g(4)) = 174$.
5. Final Answer
Thus, the value of the composite function $f(g(4))$ is 174.
### Examples
Composite functions are useful in many real-world scenarios. For instance, consider a store offering a discount. If the price of an item is represented by $g(x)$ and the discount function is $f(x)$, then $f(g(x))$ represents the final price after the discount is applied. Understanding composite functions helps in calculating the overall effect of multiple operations or transformations.
- Then, substitute $g(4)$ into $f(x)$: $f(g(4)) = f(8) = 3(8)^2 - 3(8) + 6$.
- Calculate $f(8)$: $f(8) = 3(64) - 24 + 6 = 192 - 24 + 6 = 174$.
- Therefore, $f(g(4)) = \boxed{174}$.
### Explanation
1. Understanding the Problem
We are given two functions: $f(x) = 3x^2 - 3x + 6$ and $g(x) = 2x$. Our goal is to find the value of the composite function $f(g(4))$. This means we first need to evaluate $g(4)$, and then substitute that result into the function $f(x)$.
2. Evaluating g(4)
First, let's find $g(4)$. We substitute $x=4$ into the expression for $g(x)$: $$g(4) = 2 \times 4 = 8$$
3. Substituting into f(x)
Now that we have $g(4) = 8$, we can find $f(g(4))$ by substituting $8$ into the function $f(x)$:$$f(g(4)) = f(8) = 3(8)^2 - 3(8) + 6$$
4. Calculating f(8)
Let's calculate $f(8)$:$$f(8) = 3(64) - 24 + 6 = 192 - 24 + 6 = 168 + 6 = 174$$Therefore, $f(g(4)) = 174$.
5. Final Answer
Thus, the value of the composite function $f(g(4))$ is 174.
### Examples
Composite functions are useful in many real-world scenarios. For instance, consider a store offering a discount. If the price of an item is represented by $g(x)$ and the discount function is $f(x)$, then $f(g(x))$ represents the final price after the discount is applied. Understanding composite functions helps in calculating the overall effect of multiple operations or transformations.
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