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If [tex]f(x) = x + 3[/tex], find [tex]\frac{f(x + h)}{h}[/tex].

Sagot :

Certainly! Let's solve this step-by-step.

### Given
We are given the function [tex]\( f(x) = x + 3 \)[/tex].

### Goal
We need to find the expression for [tex]\(\frac{f(x+h)}{h}\)[/tex].

1. Substitute [tex]\(x + h\)[/tex] into the function [tex]\(f\)[/tex]:
The function [tex]\( f(x) \)[/tex] is given by [tex]\( f(x) = x + 3 \)[/tex].

So, if we replace [tex]\(x\)[/tex] with [tex]\( x + h \)[/tex] in [tex]\( f(x) \)[/tex], we get:
[tex]\[ f(x + h) = (x + h) + 3 \][/tex]

2. Simplify [tex]\( f(x + h) \)[/tex]:
Simplifying the expression, we obtain:
[tex]\[ f(x + h) = x + h + 3 \][/tex]

3. Form the expression [tex]\(\frac{f(x + h)}{h}\)[/tex]:
We need to divide [tex]\( f(x + h) \)[/tex] by [tex]\( h \)[/tex]. Plugging in what we got from the previous step, we have:
[tex]\[ \frac{f(x + h)}{h} = \frac{x + h + 3}{h} \][/tex]

4. Simplify the expression:
The fraction [tex]\(\frac{x + h + 3}{h}\)[/tex] cannot be simplified further in a general algebraic form, so the final expression is:
[tex]\[ \frac{f(x + h)}{h} = \frac{x + h + 3}{h} \][/tex]

Thus, the expression for [tex]\(\frac{f(x+h)}{h}\)[/tex] is:

[tex]\[ \frac{x + h + 3}{h} \][/tex]

This concludes our solution.