Question

For functions $f\left(x\right)=x^2-13x+36$and $g\left(x\right)=x-4$.

Find$\left(a\right)\left(\frac{f}{g}\right)\left(x\right)$and$\left(b\right)\left(\frac{f}{g}\right)\left(-1\right)$.

Short answer

Part (a). $\left(\frac{f}{g}\right)\left(x\right)=x-9$

Part (b).$\left(\frac{f}{g}\right)\left(-1\right)=-10$

Step-by-step solution

Part (a). Step 1. Given information.

The function$f\left(x\right)=x^2-13x+36$and$g\left(x\right)=x-4$.

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

$$\begin{gathered} \left(\frac{f}{g}\right)\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)} \\ \Rightarrow \frac{x^2-13x+36}{x-4} \\ \Rightarrow \frac{\left(x-9\right)\left(x-4\right)}{\left(x-4\right)} \\ \Rightarrow \left(x-9\right) \end{gathered}$$

Part (a). Step 3. Simplified answer.

Hence, the obtained result is$\left(\frac{f}{g}\right)\left(x\right)=x-9$.

Part (b). Step 1. Given information.

The function$f\left(x\right)=x^2-13x+36$and$g\left(x\right)=x-4$.

Part (b). Step 2. To find the value of fg-1 by substituting x=-1 in fgx=x-9.

From the above part (a), we have $\left(\frac{f}{g}\right)\left(x\right)=x-9$.

Substitute $x=-1$in the above result.

$$\begin{gathered} \left(\frac{f}{g}\right)\left(-1\right)=-1-9 \\ \Rightarrow -10 \end{gathered}$$

Part (b). Step 3. Simplified answer.

Hence, the simplified expression is$-10$.