Let ​f(x)equals=x Superscript 6x6​, and ​g(x)equals=x cubed minus 2 x squared plus x minus 4x3−2x2+x−4.
​(a)
Find​ f(g(x)).
​(b)
Find​ g(f(x)).

To find \( f(g(x)) \) and \( g(f(x)) \), let's go step by step.

### Part (a) Find \( f(g(x)) \)

We are given:
\[
f(x) = x^6 \quad \text{and} \quad g(x) = x^3 - 2x^2 + x - 4
\]
To find \( f(g(x)) \), we substitute \( g(x) \) into \( f(x) \). Since \( f(x) = x^6 \), this means:
\[
f(g(x)) = (g(x))^6 = (x^3 - 2x^2 + x - 4)^6
\]
Thus, \( f(g(x)) \) is:
\[
f(g(x)) = (x^3 - 2x^2 + x - 4)^6
\]
Expanding this expression fully would give a very large polynomial, but this is the simplified form.

### Part (b) Find \( g(f(x)) \)

Now, to find \( g(f(x)) \), we substitute \( f(x) \) into \( g(x) \). That is:
\[
g(f(x)) = g(x^6) = (x^6)^3 - 2(x^6)^2 + x^6 - 4
\]
Simplifying each term:
\[
g(f(x)) = x^{18} - 2x^{12} + x^6 - 4
\]
So, \( g(f(x)) \) is:
\[
g(f(x)) = x^{18} - 2x^{12} + x^6 - 4
\]

### Summary:
- \( f(g(x)) = (x^3 - 2x^2 + x - 4)^6 \)
- \( g(f(x)) = x^{18} - 2x^{12} + x^6 - 4 \)

Would you like more details or have any further questions?

Here are 5 related questions:
1. How would you find the derivative of \( f(g(x)) \)?
2. Can \( f(g(x)) \) be simplified further under specific conditions?
3. How do you find the zeros of \( g(f(x)) \)?
4. What is the general strategy for computing compositions of functions?
5. How do the degrees of \( f(g(x)) \) and \( g(f(x)) \) compare?

**Tip:** When dealing with function compositions, always substitute carefully and watch for opportunities to simplify!

Let ​f(x) = x^6​, and ​g(x) = x^3 - 2x^2 + x - 4. (a) Find​ f(g(x)). (b) Find​ g(f(x)).

Learn how to solve a function composition problem involving f(x) = x^6 and g(x) = x^3 - 2x^2 + x - 4. This guide provides detailed steps to find f(g(x)) and g(f(x)) and explores key algebraic concepts like function composition and exponent laws. Suitable for high school students in Grades 10-12.

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