Finding the Composition of Functions
To find the composition of the functions f(x) = 9x^2 and g(x) = x^3, we follow these steps:
- Substitute g(x) into f(x): We need to compute f(g(x)), which means replacing every instance of x in f(x) with g(x).
- Given that g(x) = x^3, we can write:
- f(g(x)) = f(x^3)
- Substituting g(x) into f(x):
- f(x^3) = 9(x^3)^2
- Now, we simplify this further:
- f(x^3) = 9x^{6}.
Calculating for Values
The next step is to calculate specific instances as given in the problem:
- f(g(x)) for the provided expressions:
- Let’s express it in the format you have mentioned: 9x^2 + 9x^5 + 9x^{25} + 9x^{27}.
- Each term represents a different value:
- For 9x^2, we simply have the original function f(x) = 9x^2 for x = 1. This yields:
- Result: 9(1)^2 = 9
- For 9x^5, we need to evaluate g(x) with x = 1 again:
- g(1) = (1)^3 = 1 leading to f(g(1)) = f(1) = 9(1)^2 = 9
- For 9x^{25}, using the same substitution:
- g(1) leads to f(g(1)) = 9(1)^{2} = 9
- For 9x^{27}, we evaluate:
- g(1) again gives f(g(1)) = 9
Conclusion
In summary, the function compositions remain consistent across our calculations:
- f(g(x)) = 9x^{6}
- For the specific values: 9x^2, 9x^5, 9x^{25}, and 9x^{27}, the simplified results all yield the same consistent outcome: 9.