Question

if \\(f(x) = x^2 + 3\\), then \\(f(x + 2) =\\)

(a) \\(x^2 + 4x + 7\\)
(b) \\(x^2 + 2x + 7\\)
(c) \\(x^2 + x + 5\\)
(d) \\(x^2 + 7\\)
(e) \\(x^2 + 5\\)

Explanation:

Identify the given function

We are given the function:

$$f(x) = x^2 + 3$$

Substitute the new input

To find \(f(x+2)\), we replace every instance of \(x\) in the original function definition with the expression \((x+2)\):

$$f(x+2) = (x+2)^2 + 3$$

Expand the squared binomial

We expand the squared term \((x+2)^2\) using the perfect square trinomial formula:

$$(x+2)^2 = x^2 + 4x + 4$$

Simplify the expression

Now, we add the constant term \(3\) to our expanded expression:

$$f(x+2) = (x^2 + 4x + 4) + 3$$

$$f(x+2) = x^2 + 4x + 7$$

Answer:

• (A) \(x^2+4x+7\) (Correct answer)
• (B) \(x^2+2x+7\)
• (C) \(x^2+x+5\)
• (D) \(x^2+7\)
• (E) \(x^2+5\)