Question

For functions $f\left(x\right)=x^2+4x-3$and $g\left(x\right)=x^2+2x+4$.

Find (a)$\left(f·g\right)\left(x\right)$and (b)$\left(f·g\right)\left(1\right)$.

Short answer

Part (a). The solution is$x^4+6x^3+9x^2+10x-12$.

Part (b). The solution is$14$.

Step-by-step solution

Part (a). Step 1. Given information.

The function$f\left(x\right)=x^2+4x-3$and$g\left(x\right)=x^2+2x+4$.

Part (a). Step 2. To multiply the polynomial function, by substituting the value of fx and gx.

Solve it as follows:

$$\begin{gathered} \left(f·g\right)\left(x\right)=f\left(x\right)·g\left(x\right) \\ \Rightarrow \left(x^2+4x-3\right)·\left(x^2+2x+4\right) \\ \Rightarrow x^2\left(x^2+2x+4\right)+4x\left(x^2+2x+4\right)-3\left(x^2+2x+4\right) \\ \Rightarrow x^4+2x^3+4x^2+4x^3+8x^2+16x-3x^2-6x-12 \\ \Rightarrow x^4+6x^3+9x^2+10x-12 \end{gathered}$$

Part (a). Step 3. Simplified answer.

Hence, the obtained result is$x^4+6x^3+9x^2+10x-12$.

Part (b). Step 1. Given information.

The function$f\left(x\right)=x^2+4x-3$and$g\left(x\right)=x^2+2x+4$.

Part (b). Step 2. To find the value of f·g1 by substituting the value of x=1 in f·gx=x4+6x3+9x2+10x-12.

Solve it as follows:

$$\begin{gathered} \left(f·g\right)\left(1\right)={\left(1\right)}^4+6{\left(1\right)}^3+9{\left(1\right)}^2+10\left(1\right)-12 \\ \Rightarrow 1+6+9+10-12 \\ \Rightarrow 14 \end{gathered}$$

Part (b). Simplified answer.

Hence, the simplified expression is$14$.