Let's solve the problem step by step.
We are given the functions:
[tex]\[ g(x) = x - 2 \][/tex]
[tex]\[ h(x) = 4x + 4 \][/tex]
### Step 1: Find [tex]\((g + h)(x)\)[/tex]
First, let's find the expression for [tex]\( (g + h)(x) \)[/tex].
[tex]\[
(g + h)(x) = g(x) + h(x)
\][/tex]
Substitute the given functions:
[tex]\[
(g + h)(x) = (x - 2) + (4x + 4)
\][/tex]
Combine like terms:
[tex]\[
(g + h)(x) = x - 2 + 4x + 4
\][/tex]
[tex]\[
(g + h)(x) = 5x + 2
\][/tex]
So, the expression for [tex]\( (g + h)(x) \)[/tex] is:
[tex]\[
(g + h)(x) = 5x + 2
\][/tex]
### Step 2: Find [tex]\((g \cdot h)(x)\)[/tex]
Next, let's find the expression for [tex]\( (g \cdot h)(x) \)[/tex].
[tex]\[
(g \cdot h)(x) = g(x) \cdot h(x)
\][/tex]
Substitute the given functions:
[tex]\[
(g \cdot h)(x) = (x - 2) \cdot (4x + 4)
\][/tex]
Expand the expression by distributing:
[tex]\[
(g \cdot h)(x) = x \cdot (4x + 4) - 2 \cdot (4x + 4)
\][/tex]
[tex]\[
(g \cdot h)(x) = 4x^2 + 4x - 8x - 8
\][/tex]
[tex]\[
(g \cdot h)(x) = 4x^2 - 4x - 8
\][/tex]
So, the expression for [tex]\( (g \cdot h)(x) \)[/tex] is:
[tex]\[
(g \cdot h)(x) = 4x^2 - 4x - 8
\][/tex]
### Step 3: Evaluate [tex]\((g - h)(-1)\)[/tex]
Finally, let's evaluate [tex]\((g - h)(-1)\)[/tex].
First, find [tex]\( g(-1) \)[/tex] and [tex]\( h(-1) \)[/tex]:
[tex]\[
g(-1) = -1 - 2 = -3
\][/tex]
[tex]\[
h(-1) = 4(-1) + 4 = -4 + 4 = 0
\][/tex]
Now, compute [tex]\( (g - h)(-1) \)[/tex]:
[tex]\[
(g - h)(-1) = g(-1) - h(-1)
\][/tex]
[tex]\[
(g - h)(-1) = -3 - 0
\][/tex]
[tex]\[
(g - h)(-1) = -3
\][/tex]
### Summary
- The expression for [tex]\( (g + h)(x) \)[/tex] is:
[tex]\[ (g + h)(x) = 5x + 2 \][/tex]
- The expression for [tex]\( (g \cdot h)(x) \)[/tex] is:
[tex]\[ (g \cdot h)(x) = 4x^2 - 4x - 8 \][/tex]
- The value of [tex]\( (g - h)(-1) \)[/tex] is:
[tex]\[ (g - h)(-1) = -3 \][/tex]
