Burgess Frank
02/24/2023 · Middle School
ress your answer as a polynomial in standard form. \[ \begin{array}{c}f(x)=x^{2}-x-1 \\ g(x)=-x-4 \\ \text { Find: } f(g(x))\end{array} \]
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Tutor-Verified Answer
Step-by-step Solution
To find \( f(g(x)) \), we first need to substitute \( g(x) \) into \( f(x) \).
Given:
\[
f(x) = x^2 - x - 1
\]
\[
g(x) = -x - 4
\]
Now, we will substitute \( g(x) \) into \( f(x) \):
\[
f(g(x)) = f(-x - 4)
\]
Next, we replace \( x \) in \( f(x) \) with \( -x - 4 \):
\[
f(-x - 4) = (-x - 4)^2 - (-x - 4) - 1
\]
Now, we will calculate \( (-x - 4)^2 \):
\[
(-x - 4)^2 = x^2 + 8x + 16
\]
Now, substituting this back into the expression for \( f(-x - 4) \):
\[
f(-x - 4) = (x^2 + 8x + 16) - (-x - 4) - 1
\]
Now, simplify the expression:
\[
= x^2 + 8x + 16 + x + 4 - 1
\]
\[
= x^2 + 8x + x + 16 + 4 - 1
\]
\[
= x^2 + 9x + 19
\]
Thus, the polynomial \( f(g(x)) \) in standard form is:
\[
\boxed{x^2 + 9x + 19}
\]
Quick Answer
\( f(g(x)) = x^2 + 9x + 19 \)
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