Questions: Given that f(x)=x^2-20 and g(x)=8x+1, find (fg)(-1/8), if it exists. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. (fg)(-1/8)= □ (Simplify your answer.) B. The value for (fg)(-1/8) does not exist.

Given that f(x)=x^2-20 and g(x)=8x+1, find (fg)(-1/8), if it exists.

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. (fg)(-1/8)= □ (Simplify your answer.)
B. The value for (fg)(-1/8) does not exist.

Transcript text: Given that $f(x)=x^{2}-20$ and $g(x)=8 x+1$, find $(f g)\left(-\frac{1}{8}\right)$, if it exists. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. $(\mathrm{fg})\left(-\frac{1}{8}\right)=$ $\square$ (Simplify your answer.) B. The value for $(\mathrm{fg})\left(-\frac{1}{8}\right)$ does not exist.

Solution

Solution Steps

To find $(fg)\left(-\frac{1}{8}\right)$, we need to evaluate the functions $f(x)$ and $g(x)$ at $x = -\frac{1}{8}$ and then multiply the results. First, substitute $-\frac{1}{8}$ into $f(x) = x^2 - 20$ to find $f\left(-\frac{1}{8}\right)$. Next, substitute $-\frac{1}{8}$ into $g(x) = 8x + 1$ to find $g\left(-\frac{1}{8}\right)$. Finally, multiply these two results to get $(fg)\left(-\frac{1}{8}\right)$.

Step 1: Evaluate $ f\left(-\frac{1}{8}\right) $

To find $ f\left(-\frac{1}{8}\right) $, substitute $-\frac{1}{8}$ into the function $ f(x) = x^2 - 20 $: \[ f\left(-\frac{1}{8}\right) = \left(-\frac{1}{8}\right)^2 - 20 = \frac{1}{64} - 20 = -19.984375 \]

Step 2: Evaluate $ g\left(-\frac{1}{8}\right) $

To find $ g\left(-\frac{1}{8}\right) $, substitute $-\frac{1}{8}$ into the function $ g(x) = 8x + 1 $: \[ g\left(-\frac{1}{8}\right) = 8\left(-\frac{1}{8}\right) + 1 = -1 + 1 = 0 \]

Step 3: Calculate $(fg)\left(-\frac{1}{8}\right)$

Multiply the results from Step 1 and Step 2: \[ (fg)\left(-\frac{1}{8}\right) = f\left(-\frac{1}{8}\right) \cdot g\left(-\frac{1}{8}\right) = -19.984375 \cdot 0 = 0 \]