Suppose  f(x)=5x+16  and g(x)=x^2+2x−1. Find g(g(x))

To find \( g(g(x)) \), we need to substitute \( g(x) \) into itself. The given function for \( g(x) \) is:

\[
g(x) = x^2 + 2x - 1
\]

Now, let's substitute this expression into \( g(x) \) again. So, we are finding:

\[
g(g(x)) = g(x^2 + 2x - 1)
\]

Since \( g(x) = x^2 + 2x - 1 \), we replace \( x \) in this formula with \( x^2 + 2x - 1 \):

\[
g(g(x)) = (x^2 + 2x - 1)^2 + 2(x^2 + 2x - 1) - 1
\]

Let's simplify this step by step:

1. **Square \( (x^2 + 2x - 1) \):**

\[
(x^2 + 2x - 1)^2 = (x^2 + 2x - 1)(x^2 + 2x - 1)
\]

Expanding this:

\[
= x^4 + 2x^3 - x^2 + 2x^3 + 4x^2 - 2x - x^2 - 2x + 1
\]

Combining like terms:

\[
= x^4 + 4x^3 + 2x^2 - 4x + 1
\]

2. **Multiply \( 2(x^2 + 2x - 1) \):**

\[
2(x^2 + 2x - 1) = 2x^2 + 4x - 2
\]

3. **Substitute and combine all terms:**

\[
g(g(x)) = (x^4 + 4x^3 + 2x^2 - 4x + 1) + (2x^2 + 4x - 2) - 1
\]

Now, combine like terms:

\[
g(g(x)) = x^4 + 4x^3 + (2x^2 + 2x^2) + (-4x + 4x) + (1 - 2 - 1)
\]

Simplifying:

\[
g(g(x)) = x^4 + 4x^3 + 4x^2 + 0x - 2
\]

Thus, the final expression is:

\[
g(g(x)) = x^4 + 4x^3 + 4x^2 - 2
\]

### Do you have any questions or would you like further details?

Here are 5 related questions for you to explore:

1. How would you find \( f(f(x)) \) given \( f(x) = 5x + 16 \)?
2. What is the composition \( f(g(x)) \) for the same functions?
3. What is the derivative of \( g(g(x)) \)?
4. What would be \( g^{-1}(x) \), the inverse of \( g(x) \)?
5. How do you solve \( g(g(x)) = 0 \)?

**Tip:** When working with composition of functions, it's essential to substitute carefully and combine like terms step by step to avoid mistakes.

Suppose f(x)=5x+16 and g(x)=x^2+2x−1. Find g(g(x))

Explore how to find g(g(x)) for the function g(x) = x^2 + 2x - 1 through step-by-step substitution and simplification. This detailed solution walks through the expansion and simplification of polynomials, focusing on function composition. Suitable for high school students in Grades 9-12.

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