To find the expression equivalent to [tex]\((k \cdot h)(x)\)[/tex], we need to understand what it means to combine the functions [tex]\(h(x)\)[/tex] and [tex]\(k(x)\)[/tex] through multiplication.
Given:
[tex]\[ h(x) = 5 + x \][/tex]
[tex]\[ k(x) = \frac{1}{x} \][/tex]
We are asked to find [tex]\((k \cdot h)(x)\)[/tex], which represents the product of [tex]\(k(x)\)[/tex] and [tex]\(h(x)\)[/tex].
Let's break down the multiplication step by step:
1. Write down the expressions for [tex]\(h(x)\)[/tex] and [tex]\(k(x)\)[/tex]:
[tex]\[ h(x) = 5 + x \][/tex]
[tex]\[ k(x) = \frac{1}{x} \][/tex]
2. Multiply the expressions [tex]\(k(x)\)[/tex] and [tex]\(h(x)\)[/tex]:
[tex]\[ (k \cdot h)(x) = k(x) \cdot h(x) \][/tex]
3. Substitute the expressions for [tex]\(k(x)\)[/tex] and [tex]\(h(x)\)[/tex]:
[tex]\[ (k \cdot h)(x) = \left(\frac{1}{x}\right) \cdot (5 + x) \][/tex]
4. Perform the multiplication:
[tex]\[ (k \cdot h)(x) = \frac{1}{x} \cdot (5 + x) \][/tex]
5. Distribute [tex]\(\frac{1}{x}\)[/tex] across [tex]\(5 + x\)[/tex]:
[tex]\[ (k \cdot h)(x) = \frac{1}{x} \cdot 5 + \frac{1}{x} \cdot x \][/tex]
6. Simplify:
[tex]\[ (k \cdot h)(x) = \frac{5}{x} + \frac{x}{x} \][/tex]
[tex]\[ (k \cdot h)(x) = \frac{5}{x} + 1 \][/tex]
Combining the fractions:
[tex]\[ (k \cdot h)(x) = \frac{5 + x}{x} \][/tex]
Thus, the expression equivalent to [tex]\((k \cdot h)(x)\)[/tex] is:
[tex]\[ \frac{5 + x}{x} \][/tex]
Therefore, the correct answer is:
[tex]\[
\frac{(5+x)}{x}
\][/tex]
