Q.324. - Intermediate Algebra

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

Step-by-Step Solution

Answer

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

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

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 f g x by substituting the value of f x and g x .

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

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 f g - 1 by substituting x = - 1 in f g x = 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 $-10$.