Question

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Complete this activity. Include all of your work in your final answer. Submit your solution.

Given: f(x) = x^2 + 2x + 1, find f(x + h) and simplify.

Answer 1

Explanation:

To find f(x + h), we substitute x + h for x in the function f(x):

f(x + h) = (x + h)^2 + 2(x + h) + 1

Now we simplify by expanding the square:

f(x + h) = x^2 + 2hx + h^2 + 2x + 2h + 1

We can combine like terms to simplify this expression:

f(x + h) = x^2 + (2h + 2)x + (h^2 + 2h + 1)

Therefore, f(x + h) = x^2 + (2h + 2)x + (h^2 + 2h + 1).

Answer 2

`f(x+h) = x^2 + 2xh + h^2 + 2x+2h + 1`

Explanation:

In this question, we have to plug (x + h) into the given function and simplify.

`f(x) = x^2 + 2x + 1`

↓ replacing all instances of x with (x + h)

`f(x+h) = (x+h)^2 + 2(x+h) + 1`

↓ expanding the quadratic term (x + h)²

`f(x+h) = (x^2 + 2xh + h^2) + 2(x+h) + 1`

↓ applying the distributive property ...
`2(x+h) = 2x + 2h`

`\boxed{f(x+h) = x^2 + 2xh + h^2 + 2x+2h + 1}`

No further simplification can be done.