Given the following functions  
f
(
x
)
  and  
g
(
x
)
 , find  
(
f
g
)
(
x
)
 .
f
(
x
)
=
√
x
g
(
x
)
=
1
+
4/x

To find \((f \cdot g)(x)\), you need to multiply the two functions \(f(x)\) and \(g(x)\). The given functions are:

- \(f(x) = \sqrt{x}\)
- \(g(x) = 1 + \frac{4}{x}\)

Now, multiply these two functions:

\[
(f \cdot g)(x) = f(x) \cdot g(x)
\]

Substitute the expressions for \(f(x)\) and \(g(x)\):

\[
(f \cdot g)(x) = \sqrt{x} \cdot \left( 1 + \frac{4}{x} \right)
\]

Distribute \(\sqrt{x}\) across the terms inside the parentheses:

\[
(f \cdot g)(x) = \sqrt{x} \cdot 1 + \sqrt{x} \cdot \frac{4}{x}
\]

Simplify each term:

\[
(f \cdot g)(x) = \sqrt{x} + \frac{4\sqrt{x}}{x}
\]

Since \(\frac{\sqrt{x}}{x} = \frac{1}{\sqrt{x}}\), we can simplify further:

\[
(f \cdot g)(x) = \sqrt{x} + \frac{4}{\sqrt{x}}
\]

Thus, the product of the functions is:

\[
(f \cdot g)(x) = \sqrt{x} + \frac{4}{\sqrt{x}}
\]

Would you like further clarification on the steps?

Here are 5 related questions to practice:
1. What is the domain of the function \((f \cdot g)(x)\)?
2. How do you differentiate \((f \cdot g)(x)\)?
3. What is the value of \((f \cdot g)(4)\)?
4. Can you simplify \(\sqrt{x} + \frac{4}{\sqrt{x}}\) further using a common denominator?
5. What happens to \((f \cdot g)(x)\) as \(x\) approaches infinity?

**Tip:** When multiplying functions, always be mindful of simplifying any complex expressions, such as fractions involving square roots.

Given the following functions f(x) and g(x), find (f g)(x). f(x) = √x, g(x) = 1 + 4/x.

Explore how to multiply the functions f(x) = √x and g(x) = 1 + 4/x. This step-by-step guide covers function multiplication, simplifying radicals, and algebraic expressions. Ideal for students in Grades 9-11.

Math Problem Statement

Given the following functions
f ( x ) and
g ( x ) , find
( f g ) ( x ) . f ( x )

√ x g ( x )

Solution

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