Question
If \( f(x)=x^{2}+8 x+3 \), what is \( f(x+3) \) ? (Simplify your answer complete.............................................................................. \( f(x+3)=\square \) TUTORIAL
Ask by Horton Norris. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( f(x+3) = x^2 + 14x + 36 \)
Solution
1. The function is given by
\[
f(x) = x^2 + 8x + 3.
\]
2. To find \( f(x+3) \), substitute \( x+3 \) for \( x \) in the function:
\[
f(x+3) = (x+3)^2 + 8(x+3) + 3.
\]
3. Expand \((x+3)^2\):
\[
(x+3)^2 = x^2 + 6x + 9.
\]
4. Expand the term \(8(x+3)\):
\[
8(x+3) = 8x + 24.
\]
5. Substitute the expanded terms back into the expression:
\[
f(x+3) = x^2 + 6x + 9 + 8x + 24 + 3.
\]
6. Combine like terms:
\[
f(x+3) = x^2 + (6x + 8x) + (9 + 24 + 3).
\]
7. Simplify:
\[
f(x+3) = x^2 + 14x + 36.
\]
Thus, the simplified expression for \( f(x+3) \) is
\[
\boxed{x^2 + 14x + 36}.
\]
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Simplify this solution Beyond the Answer
To find \( f(x+3) \), we need to substitute \( x+3 \) into the function \( f(x) = x^2 + 8x + 3 \). Starting with the substitution: \[ f(x+3) = (x+3)^2 + 8(x+3) + 3 \] Now, we simplify each term: 1. \( (x+3)^2 = x^2 + 6x + 9 \) 2. \( 8(x+3) = 8x + 24 \) Now, putting it all together: \[ f(x+3) = (x^2 + 6x + 9) + (8x + 24) + 3 \] Combining like terms: \[ f(x+3) = x^2 + (6x + 8x) + (9 + 24 + 3) \] \[ f(x+3) = x^2 + 14x + 36 \] So the simplified answer is: \[ f(x+3) = x^2 + 14x + 36 \]
