To solve for [tex]\( f(a + h) - f(a) \)[/tex] where the function [tex]\( f(x) = 3x + 1 \)[/tex], let's follow these steps:
1. Compute [tex]\( f(a + h) \)[/tex]:
[tex]\[
f(a + h) = 3(a + h) + 1
\][/tex]
Expanding this, we get:
[tex]\[
f(a + h) = 3a + 3h + 1
\][/tex]
2. Compute [tex]\( f(a) \)[/tex]:
[tex]\[
f(a) = 3a + 1
\][/tex]
3. Find the difference [tex]\( f(a + h) - f(a) \)[/tex]:
[tex]\[
f(a + h) - f(a) = (3a + 3h + 1) - (3a + 1)
\][/tex]
Simplifying this expression, we notice that the [tex]\( 3a \)[/tex] and [tex]\( +1 \)[/tex] terms cancel out:
[tex]\[
f(a + h) - f(a) = 3a + 3h + 1 - 3a - 1 = 3h
\][/tex]
Hence, the difference [tex]\( f(a + h) - f(a) \)[/tex] simplifies to:
[tex]\[
3h
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{3h}
\][/tex]
