Question

For functions $f\left(x\right)=x^2-15x+45$and $g\left(x\right)=x-9$.

Find (a)$\left(\frac{f}{g}\right)\left(x\right)$and (b)$\left(\frac{f}{g}\right)(-5)$.

Short answer

Part (a). The solution is $\left(x-6\right)$.

Part (b). The solution is$-11$.

Step-by-step solution

Part (a). Step 1. Given information.

The functions$f\left(x\right)=x^2-15x+54$and$g\left(x\right)=x-9$.

Part (a). Step 2. Simplify fgx by substituting the value of fx and gx.

Solve it as follows:

$$\begin{gathered} \left(\frac{f}{g}\right)\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)} \\ \Rightarrow \frac{x^2-15x+54}{x-9} \\ \Rightarrow \frac{\left(x-9\right)\left(x-6\right)}{\left(x-9\right)} \\ \Rightarrow \left(x-6\right) \end{gathered}$$

Part (a). Step 3. Simplified answer.

Hence, the obtained result is$\left(\frac{f}{g}\right)\left(x\right)=\left(x-6\right)$.

Part (b). Step 1. Given information.

The functions$f\left(x\right)=x^2-15x+54$and$g\left(x\right)=x-9$.

Part (b). To find the value of fg-5 by substituting x=-5.

From the above part (a), we have $\left(\frac{f}{g}\right)\left(x\right)=x-6$.

Substitute $x=-5$in the above result.

$$\begin{gathered} \left(\frac{f}{g}\right)\left(-5\right)=-5-6 \\ \Rightarrow -11 \end{gathered}$$

Part (b). Step 3. Simplified answer.

Hence, the simplified expression is$-11$.