Questions: Given that f(x)=x^2-11 and g(x)=3x+1, find (f-g)(-3), if it exists. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. (f-g)(-3)= □ (Simplify your answer.) B. The value for (f-g)(-3) does not exist.

Given that f(x)=x^2-11 and g(x)=3x+1, find (f-g)(-3), if it exists.

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. (f-g)(-3)= □ (Simplify your answer.)
B. The value for (f-g)(-3) does not exist.

Transcript text: Given that $f(x)=x^{2}-11$ and $g(x)=3 x+1$, find $(f-g)(-3)$, if it exists. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. $(f-g)(-3)=$ $\square$ (Simplify your answer.) B. The value for $(\mathrm{f}-\mathrm{g})(-3)$ does not exist.

Solution

Solution Steps

To solve the problem, we need to find the value of \((f-g)(-3)\). This involves the following steps:

Evaluate \(f(-3)\) by substituting \(-3\) into the function \(f(x)\).
Evaluate \(g(-3)\) by substituting \(-3\) into the function \(g(x)\).
Subtract the value of \(g(-3)\) from \(f(-3)\) to get \((f-g)(-3)\).

Step 1: Evaluate \( f(-3) \)

To find \( f(-3) \), we substitute \(-3\) into the function \( f(x) = x^2 - 11 \): \[ f(-3) = (-3)^2 - 11 = 9 - 11 = -2 \]

Step 2: Evaluate \( g(-3) \)

Next, we evaluate \( g(-3) \) by substituting \(-3\) into the function \( g(x) = 3x + 1 \): \[ g(-3) = 3(-3) + 1 = -9 + 1 = -8 \]

Step 3: Calculate \( (f-g)(-3) \)

Now, we find \( (f-g)(-3) \) by subtracting \( g(-3) \) from \( f(-3) \): \[ (f-g)(-3) = f(-3) - g(-3) = -2 - (-8) = -2 + 8 = 6 \]

Final Answer

The value of \( (f-g)(-3) \) is \(\boxed{6}\).

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