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Functions for reference:
- [tex]\( f(x) = x + 5 \)[/tex]
- [tex]\( g(x) = x^2 + 6x + 5 \)[/tex]
- [tex]\( h(x) = 2x - 3 \)[/tex]
- [tex]\( j(x) = -4x \)[/tex]

Using the functions given above, find [tex]\((h-f)(x)\)[/tex].

[tex]\[
(h-f)(x) = h(x) - f(x)
\][/tex]
[tex]\[
(h-f)(x) = (2x - 3) - (x + 5)
\][/tex]
[tex]\[
(h-f)(x) = 2x - 3 - x - 5
\][/tex]
[tex]\[
(h-f)(x) = x - 8
\][/tex]

Now, let's try another. Find [tex]\( m(x) = (g-j)(x) \)[/tex].

[tex]\[
m(x) = g(x) - j(x)
\][/tex]
[tex]\[
m(x) = x^2 + 6x + 5 - (-4x)
\][/tex]
[tex]\[
m(x) = x^2 + 6x + 5 + 4x
\][/tex]
[tex]\[
m(x) = x^2 + 10x + 5
\][/tex]

Evaluate [tex]\( m(3) \)[/tex].

[tex]\[
\begin{array}{l}
m(3) = (3)^2 + 10(3) + 5 \\
m(3) = 9 + 30 + 5 \\
m(3) = 44
\end{array}
\][/tex]


Sagot :

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].