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Find and simplify each of the following for f(x)=7x-6.
(A) f(x + h)
(B) f(x+h) f(x)
f(x + h) - f(x)
(C)
h
(A) f(x+h) = (Do not factor.)

Sagot :

GinnyAnsweridn
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.
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