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.
