Let's carefully go through the process of finding [tex]\((h - f)(x)\)[/tex] and [tex]\(m(x) = (g - j)(x)\)[/tex], and then evaluating [tex]\(m(3)\)[/tex].
### Find [tex]\((h - f)(x)\)[/tex]:
Given:
- [tex]\( f(x) = x + 5 \)[/tex]
- [tex]\( h(x) = 2x - 3 \)[/tex]
Step-by-step process:
1. Write the expression for [tex]\((h - f)(x)\)[/tex]:
[tex]\[
(h - f)(x) = h(x) - f(x) = (2x - 3) - (x + 5)
\][/tex]
2. Distribute the subtraction to each term:
[tex]\[
(h - f)(x) = 2x - 3 - x - 5
\][/tex]
3. Combine like terms:
[tex]\[
2x - x - 3 - 5 = x - 8
\][/tex]
So, [tex]\((h - f)(x) = x - 8\)[/tex].
### Find [tex]\(m(x) = (g - j)(x)\)[/tex]:
Given:
- [tex]\( g(x) = x^2 + 6x + 5 \)[/tex]
- [tex]\( j(x) = -4x \)[/tex]
Step-by-step process:
1. Write the expression for [tex]\((g - j)(x)\)[/tex]:
[tex]\[
m(x) = (g - j)(x) = g(x) - j(x) = (x^2 + 6x + 5) - (-4x)
\][/tex]
2. Distribute the subtraction to each term, noting that subtracting a negative is the same as adding:
[tex]\[
m(x) = x^2 + 6x + 5 + 4x
\][/tex]
3. Combine like terms:
[tex]\[
x^2 + 6x + 4x + 5 = x^2 + 10x + 5
\][/tex]
So, [tex]\(m(x) = x^2 + 10x + 5\)[/tex].
### Evaluate [tex]\(m(3)\)[/tex]:
Given [tex]\(m(x) = x^2 + 10x + 5\)[/tex]:
Substitute [tex]\(x = 3\)[/tex] into the function:
[tex]\[
m(3) = (3)^2 + 10(3) + 5
\][/tex]
[tex]\[
m(3) = 9 + 30 + 5
\][/tex]
[tex]\[
m(3) = 44
\][/tex]
Therefore, [tex]\(m(3) = 44\)[/tex].
