For the given functions f(x) and g(x), find f(x) ∙ g(x).
-$$f ( x ) = \frac { x ^ { 2 } + 13 x + 42 } { x ^ { 2 } + 14 x + 49 } , g ( x ) = \frac { x ^ { 2 } + 7 x } { x ^ { 2 } - 3 x - 54 }$$
A) $\frac { 1 } { x - 9 }$

B) $\frac { x } { x ^ { 2 } + 14 x + 49 }$

C) $\frac { x } { x - 9 }$

D) $\frac { x ^ { 2 } + 7 x } { x - 9 }$

Question

For the given functions f(x) and g(x), find f(x) ∙ g(x).
-$$f ( x ) = \frac { x ^ { 2 } + 13 x + 42 } { x ^ { 2 } + 14 x + 49 } , g ( x ) = \frac { x ^ { 2 } + 7 x } { x ^ { 2 } - 3 x - 54 }$$
A) $\frac { 1 } { x - 9 }$

B) $\frac { x } { x ^ { 2 } + 14 x + 49 }$

C) $\frac { x } { x - 9 }$

D) $\frac { x ^ { 2 } + 7 x } { x - 9 }$

Quizplus · Accepted Answer

To find $f(x) \cdot g(x)$, we multiply the two given functions:$$f(x) = \frac{x^2 + 13x + 42}{x^2 + 14x + 49},$$$$g(x) = \frac{x^2 + 7x}{x^2 - 3x - 54}.$$Multiplying $f(x)$ and $g(x)$ gives:$$f(x) \cdot g(x) = \left(\frac{x^2 + 13x + 42}{x^2 + 14x + 49}\right) \cdot \left(\frac{x^2 + 7x}{x^2 - 3x - 54}\right).$$Simplifying the expressions:- The numerator of $f(x)$, $x^2 + 13x + 42$, can be factored as $(x + 6)(x + 7)$.- The denominator of $f(x)$, $x^2 + 14x + 49$, is $(x + 7)^2$.- The numerator of $g(x)$ is already simplified as $x(x + 7)$.- The denominator of $g(x)$, $x^2 - 3x - 54$, can be factored as $(x - 9)(x + 6)$.Substituting these factors into the multiplication gives:$$f(x) \cdot g(x) = \left(\frac{(x + 6)(x + 7)}{(x + 7)^2}\right) \cdot \left(\frac{x(x + 7)}{(x - 9)(x + 6)}\right).$$Simplifying, we see that $(x + 7)$ and $(x + 6)$ cancel out from the numerator and denominator, leaving:$$f(x) \cdot g(x) = \frac{x}{x - 9}.$$Therefore, the correct answer is $\frac{x}{x - 9}$, which is option C.

For the Given Functions F(x) and G(x), Find F(x) ∙ f(x)=x2+13x+42x2+14x+49,g(x)=x2+7xx2−3x−54f ( x ) = \frac { x ^ { 2 } + 13 x + 42 } { x ^ { 2 } + 14 x + 49 } , g ( x ) = \frac { x ^ { 2 } + 7 x } { x ^ { 2 } - 3 x - 54 }f(x)=x2+14x+49x2+13x+42​,g(x)=x2−3x−54x2+7x​

For the Given Functions F(x) and G(x), Find F(x) ∙ $f ( x ) = \frac { x ^ { 2 } + 13 x + 42 } { x ^ { 2 } + 14 x + 49 } , g ( x ) = \frac { x ^ { 2 } + 7 x } { x ^ { 2 } - 3 x - 54 }$