Certainly! Let's go through the problem step-by-step and simplify each part:
Given the function [tex]\( f(x) = 7x - 6 \)[/tex],
### (A) Compute [tex]\( f(x + h) \)[/tex]
To find [tex]\( f(x + h) \)[/tex], substitute [tex]\( x + h \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[ f(x + h) = 7(x + h) - 6 \][/tex]
[tex]\[ = 7x + 7h - 6 \][/tex]
So,
[tex]\[ f(x + h) = 7x + 7h - 6 \][/tex]
### (B) Compute [tex]\( f(x + h) - f(x) \)[/tex]
Using the expressions for [tex]\( f(x + h) \)[/tex] and [tex]\( f(x) \)[/tex]:
[tex]\[ f(x + h) - f(x) = (7x + 7h - 6) - (7x - 6) \][/tex]
Simplify the expression:
[tex]\[ = 7x + 7h - 6 - 7x + 6 \][/tex]
[tex]\[ = 7h \][/tex]
Therefore,
[tex]\[ f(x + h) - f(x) = 7h \][/tex]
### (C) Compute [tex]\(\frac{f(x + h) - f(x)}{h}\)[/tex]
Now let's find the difference quotient:
[tex]\[ \frac{f(x + h) - f(x)}{h} \][/tex]
Substitute the result from part (B) into the expression:
[tex]\[ \frac{7h}{h} \][/tex]
Simplify the fraction:
[tex]\[ = 7 \][/tex]
Thus,
[tex]\[ \frac{f(x + h) - f(x)}{h} = 7 \][/tex]
### Summary
(A) [tex]\( f(x + h) = 7x + 7h - 6 \)[/tex]
(B) [tex]\( f(x + h) - f(x) = 7h \)[/tex]
(C) [tex]\(\frac{f(x + h) - f(x)}{h} = 7 \)[/tex]
These are the simplified expressions for each part of the question.
