Question Find the polynomial equivalent to $(f \cdot g)(x)$ , where $f(x) = x + 1$ and $g(x) = \frac{2}{x}$ .

Studdy Solution

STEP 1

1. The functions $f(x)=x+1$ and $g(x)=\frac{2}{x}$ are defined for all $x$ in their respective domains, where $x \neq 0$ for $g(x)$ .
2. The notation $(f \cdot g)(x)$ represents the product of the functions $f(x)$ and $g(x)$ .
3. The product of the functions should yield a polynomial or a rational expression that can be simplified to one of the given answer choices.

STEP 2

1. Multiply the functions $f(x)$ and $g(x)$ to find $(f \cdot g)(x)$ .
2. Simplify the product to match one of the given answer choices.

STEP 3

Multiply the functions $f(x)$ and $g(x)$ to find the expression for $(f \cdot g)(x)$ .
$(f \cdot g)(x) = f(x) \cdot g(x) = (x+1) \cdot \left(\frac{2}{x}\right)$

STEP 4

Distribute the multiplication across the terms in $f(x)$ .
$(f \cdot g)(x) = x \cdot \frac{2}{x} + 1 \cdot \frac{2}{x}$

STEP 5

Simplify the expression by canceling out the $x$ terms where possible and combining like terms.
$(f \cdot g)(x) = 2 + \frac{2}{x}$

STEP 6

Recognize that the expression $2 + \frac{2}{x}$ is not a polynomial, but rather a rational expression. Rewrite the expression with a common denominator to potentially match one of the answer choices.
$(f \cdot g)(x) = \frac{2x}{x} + \frac{2}{x}$

STEP 7

Combine the fractions over the common denominator.
$(f \cdot g)(x) = \frac{2x + 2}{x}$

STEP 8

Compare the simplified expression to the answer choices to find the equivalent polynomial.
The expression $\frac{2x + 2}{x}$ matches answer choice B. $\frac{2x + 2}{x}$ .
The equivalent polynomial to $(f \cdot g)(x)$ is answer choice B. $\frac{2x + 2}{x}$ .

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