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Perform the indicated operations: \[ \begin{array}{ll} f(x)=x-3 & g(x)=x^{2}-2 ...

Aug 20, 2024

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Solution by Steps

step 1

To find

(f \cdot p)(x)

, we calculate:

(x - 3)(x - 1) = (x - 3)(x - 1) = x^2 - 4x + 3

step 2

To find

(f \cdot g)(x)

, we calculate:

(x - 3)(x^2 - 2x - 3) = (x - 3)(x - 3)(x + 1) = (x - 3)^2(x + 1) = x^3 - 5x^2 + 3x + 9

step 3

To find

\left(\frac{g}{f}\right)(x)

, we calculate:

\frac{x^2 - 2x - 3}{x - 3} = x + 1

for

x \neq 3

step 4

To find

\left(\frac{p}{v}\right)(x)

, we calculate:

\frac{x - 1}{x^2 + 5x + 4} = \frac{x - 1}{(x + 1)(x + 4)}

step 5

To find

\left(\frac{g}{v}\right)(x)

, we calculate:

\frac{x^2 - 2x - 3}{x^2 + 5x + 4} = \frac{(x + 1)(x - 3)}{(x + 1)(x + 4)} = \frac{x - 3}{x + 4}

for

x \neq -1

Answer

The results of the operations are: 1.

(f \cdot p)(x) = x^2 - 4x + 3

(f \cdot g)(x) = x^3 - 5x^2 + 3x + 9

\left(\frac{g}{f}\right)(x) = x + 1

for

x \neq 3

\left(\frac{p}{v}\right)(x) = \frac{x - 1}{(x + 1)(x + 4)}

\left(\frac{g}{v}\right)(x) = \frac{x - 3}{x + 4}

for

x \neq -1

Key Concept

Function operations including multiplication and division of polynomial functions.

Explanation

The operations involve substituting the defined functions into the expressions and simplifying them accordingly.