Question

for $f(x) = 1 - x$ and $g(x) = 4x^2 + x + 7$, find the following functions.
a. $(f \circ g)(x)$; b. $(g \circ f)(x)$; c. $(f \circ g)(3)$; d. $(g \circ f)(3)$
a. $(f \circ g)(x) = \square$
(simplify your answer.)
b. $(g \circ f)(x) = \square$
(simplify your answer.)
c. $(f \circ g)(3) = \square$
d. $(g \circ f)(3) = \square$

Explanation:

Step1: Find \((f \circ g)(x)\)

The composition \((f \circ g)(x)\) means \(f(g(x))\). We substitute \(g(x)\) into \(f(x)\). Given \(f(x)=1 - x\) and \(g(x)=4x^{2}+x + 7\), we replace \(x\) in \(f(x)\) with \(g(x)\):
\(f(g(x))=1-(4x^{2}+x + 7)\)
Simplify the expression:
\(f(g(x))=1 - 4x^{2}-x - 7=-4x^{2}-x - 6\)

Step2: Find \((g \circ f)(x)\)

The composition \((g \circ f)(x)\) means \(g(f(x))\). We substitute \(f(x)\) into \(g(x)\). Given \(f(x)=1 - x\) and \(g(x)=4x^{2}+x + 7\), we replace \(x\) in \(g(x)\) with \(f(x)\):
\(g(f(x))=4(1 - x)^{2}+(1 - x)+7\)
First, expand \((1 - x)^{2}\) using the formula \((a - b)^{2}=a^{2}-2ab + b^{2}\), where \(a = 1\) and \(b=x\):
\((1 - x)^{2}=1-2x+x^{2}\)
Then substitute back into \(g(f(x))\):
\(g(f(x))=4(1 - 2x+x^{2})+1 - x+7\)
Distribute the 4:
\(g(f(x))=4-8x + 4x^{2}+1 - x+7\)
Combine like terms:
\(4x^{2}+(-8x - x)+(4 + 1+7)=4x^{2}-9x + 12\)

Step3: Find \((f \circ g)(3)\)

We can use the function we found in Step1: \((f \circ g)(x)=-4x^{2}-x - 6\). Substitute \(x = 3\) into this function:
\((f \circ g)(3)=-4(3)^{2}-3 - 6\)
First, calculate \(3^{2}=9\):
\(-4\times9-3 - 6=-36-3 - 6=-45\)
Alternatively, we can first find \(g(3)\) and then substitute into \(f(x)\). \(g(3)=4(3)^{2}+3 + 7=4\times9+3 + 7=36 + 3+7 = 46\). Then \(f(46)=1-46=-45\).

Step4: Find \((g \circ f)(3)\)

We can use the function we found in Step2: \((g \circ f)(x)=4x^{2}-9x + 12\). Substitute \(x = 3\) into this function:
\((g \circ f)(3)=4(3)^{2}-9(3)+12\)
Calculate \(3^{2}=9\):
\(4\times9-27 + 12=36-27 + 12=21\)
Alternatively, we can first find \(f(3)\) and then substitute into \(g(x)\). \(f(3)=1 - 3=-2\). Then \(g(-2)=4(-2)^{2}+(-2)+7=4\times4-2 + 7=16-2 + 7 = 21\).

Answer:

a. \(-4x^{2}-x - 6\)
b. \(4x^{2}-9x + 12\)
c. \(-45\)
d. \(21\)