Question

suppose that the functions g and r are defined as follows.
g(x) = 2x + 2
r(x) = x² - 1
find the following.
(g ∘ r)(-1) =
(r ∘ g)(-1) =

Explanation:

Step1: Find \((g \circ r)(-1)\)

First, recall that \((g \circ r)(x)=g(r(x))\). So we need to find \(r(-1)\) first, then substitute that into \(g(x)\).

Find \(r(-1)\):
\(r(x)=x^2 - 1\), so \(r(-1)=(-1)^2 - 1\)
\(=1 - 1 = 0\)

Now substitute \(r(-1)=0\) into \(g(x)\):
\(g(x)=2x + 2\), so \(g(0)=2(0)+2 = 2\)
Thus, \((g \circ r)(-1)=2\)

Step2: Find \((r \circ g)(-1)\)

Recall that \((r \circ g)(x)=r(g(x))\). So we need to find \(g(-1)\) first, then substitute that into \(r(x)\).

Find \(g(-1)\):
\(g(x)=2x + 2\), so \(g(-1)=2(-1)+2\)
\(=-2 + 2 = 0\)

Now substitute \(g(-1)=0\) into \(r(x)\):
\(r(x)=x^2 - 1\), so \(r(0)=0^2 - 1 = -1\)
Thus, \((r \circ g)(-1)=-1\)

Answer:

\((g \circ r)(-1)=\boxed{2}\)
\((r \circ g)(-1)=\boxed{-1}\)