Question

for $f(x) = 2x + 1$ and $g(x) = x^2 - 7$, find $(f - g)(x)$. \
a. $-x^2 + 2x + 8$ \
b. $2x^2 - 15$ \
c. $x^2 - 2x - 8$ \
d. $-x^2 + 2x - 6$

Explanation:

Step1: Recall the formula for (f - g)(x)

The formula for the difference of two functions is \((f - g)(x)=f(x)-g(x)\).

Step2: Substitute the given functions

We know \(f(x) = 2x + 1\) and \(g(x)=x^{2}-7\), so substitute these into the formula:
\((f - g)(x)=(2x + 1)-(x^{2}-7)\)

Step3: Distribute the negative sign

Distribute the negative sign to each term in \(g(x)\):
\((f - g)(x)=2x + 1 - x^{2}+7\)

Step4: Combine like terms

Combine the constant terms \(1\) and \(7\):
\((f - g)(x)=-x^{2}+2x+(1 + 7)=-x^{2}+2x + 8\)

Answer:

A. \(-x^{2}+2x + 8\)