If [tex][tex]$h(x)=5+x$[/tex][/tex] and [tex][tex]$k(x)=\frac{1}{x}$[/tex][/tex], which expression is equivalent to [tex][tex]$(k \cdot h)(x)$[/tex][/tex]?

A. [tex]\frac{(5+x)}{x}[/tex]
B. [tex]\frac{1}{(5+x)}[/tex]
C. [tex]5+\left(\frac{1}{x}\right)[/tex]
D. [tex]5+(5+x)[/tex]



Answer :

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]