Q.329. - Intermediate Algebra

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) $\left(f·g\right)\left(1\right)$.

Step-by-Step Solution

Answer

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

Part (b). The solution is $14$.

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 f x and g x .

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 · g 1 by substituting the value of x = 1 in f · g x = x 4 + 6 x 3 + 9 x 2 + 10 x - 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 $14$.