To determine if [tex]\( f(x) = 2x + 2 \)[/tex] and [tex]\( g(x) = -2x - 2 \)[/tex] are inverse functions, we need to check if [tex]\( f(g(x)) = x \)[/tex] and [tex]\( g(f(x)) = x \)[/tex]. Let's go through the steps for both compositions.
### Step 1: Compute [tex]\( f(g(x)) \)[/tex]
1. Substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[
f(g(x)) = f(-2x - 2)
\][/tex]
2. Substitute [tex]\(-2x - 2\)[/tex] for [tex]\( x \)[/tex] in [tex]\( f(x) \)[/tex]:
[tex]\[
f(-2x - 2) = 2(-2x - 2) + 2
\][/tex]
3. Simplify the expression:
[tex]\[
2(-2x - 2) + 2 = -4x - 4 + 2 = -4x - 2
\][/tex]
### Step 2: Check if [tex]\( f(g(x)) = x \)[/tex]
Now, compare [tex]\( -4x - 2 \)[/tex] with [tex]\( x \)[/tex]:
[tex]\[
-4x - 2 \neq x
\][/tex]
Since [tex]\( f(g(x)) \neq x \)[/tex], [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are not inverses.
### Step 3: Compute [tex]\( g(f(x)) \)[/tex]
1. Substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[
g(f(x)) = g(2x + 2)
\][/tex]
2. Substitute [tex]\( 2x + 2 \)[/tex] for [tex]\( x \)[/tex] in [tex]\( g(x) \)[/tex]:
[tex]\[
g(2x + 2) = -2(2x + 2) - 2
\][/tex]
3. Simplify the expression:
[tex]\[
-2(2x + 2) - 2 = -4x - 4 - 2 = -4x - 6
\][/tex]
### Step 4: Check if [tex]\( g(f(x)) = x \)[/tex]
Now, compare [tex]\( -4x - 6 \)[/tex] with [tex]\( x \)[/tex]:
[tex]\[
-4x - 6 \neq x
\][/tex]
Since [tex]\( g(f(x)) \neq x \)[/tex], [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are not inverses.
Based on our calculations, [tex]\( f(g(x)) \neq x \)[/tex] and [tex]\( g(f(x)) \neq x \)[/tex]. Therefore, we conclude that:
No, [tex]\( f(x) = 2x + 2 \)[/tex] and [tex]\( g(x) = -2x - 2 \)[/tex] are not inverse functions.
