Sure, let's solve this step-by-step. We're given the function [tex]\( f(x) = 7x - 1 \)[/tex]. We need to evaluate and simplify the expressions for [tex]\( f(a) \)[/tex], [tex]\( f(a+h) \)[/tex], and [tex]\(\frac{f(a+h)-f(a)}{h} \)[/tex].
1. Evaluate [tex]\( f(a) \)[/tex]:
Substitute [tex]\( x = a \)[/tex] into the function [tex]\( f(x) = 7x - 1 \)[/tex]:
[tex]\[
f(a) = 7a - 1
\][/tex]
Therefore,
[tex]\[
f(a) = 7a - 1
\][/tex]
2. Evaluate [tex]\( f(a+h) \)[/tex]:
Substitute [tex]\( x = a + h \)[/tex] into the function [tex]\( f(x) = 7x - 1 \)[/tex]:
[tex]\[
f(a + h) = 7(a + h) - 1
\][/tex]
Simplify the expression:
[tex]\[
f(a + h) = 7a + 7h - 1
\][/tex]
Thus,
[tex]\[
f(a + h) = 7a + 7h - 1
\][/tex]
3. Simplify [tex]\(\frac{f(a+h) - f(a)}{h} \)[/tex]:
First, find the difference [tex]\( f(a+h) - f(a) \)[/tex]:
[tex]\[
f(a+h) - f(a) = (7a + 7h - 1) - (7a - 1)
\][/tex]
Simplify the numerator:
[tex]\[
f(a+h) - f(a) = 7a + 7h - 1 - 7a + 1
\][/tex]
[tex]\[
f(a+h) - f(a) = 7h
\][/tex]
Now, divide by [tex]\( h \)[/tex]:
[tex]\[
\frac{f(a+h) - f(a)}{h} = \frac{7h}{h}
\][/tex]
Simplify the expression:
[tex]\[
\frac{7h}{h} = 7
\][/tex]
Thus,
[tex]\[
\frac{f(a+h) - f(a)}{h} = 7
\][/tex]
So the simplified answers are:
[tex]\[
f(a) = 7a - 1
\][/tex]
[tex]\[
f(a+h) = 7a + 7h - 1
\][/tex]
[tex]\[
\frac{f(a+h) - f(a)}{h} = 7
\][/tex]
