Sure, let's go through this step by step:
1. First, define the function [tex]\( f(x) = x - 5 \)[/tex]:
- This is a linear function where for any input [tex]\( x \)[/tex], the output is [tex]\( x - 5 \)[/tex].
2. Find [tex]\( f(h) \)[/tex]:
- We substitute [tex]\( h \)[/tex] into the function: [tex]\( f(h) = h - 5 \)[/tex].
3. Find [tex]\( f(x + h) \)[/tex]:
- We substitute [tex]\( x + h \)[/tex] into the function: [tex]\( f(x + h) = (x + h) - 5 \)[/tex].
4. Find [tex]\( f(x + h) - f(x) \)[/tex]:
- First, find [tex]\( f(x) \)[/tex]: [tex]\( f(x) = x - 5 \)[/tex].
- Now calculate the difference:
[tex]\[
f(x + h) - f(x) = [(x + h) - 5] - [x - 5]
\][/tex]
Let's simplify this expression:
[tex]\[
f(x + h) - f(x) = (x + h - 5) - (x - 5)
\][/tex]
Distribute the negative sign:
[tex]\[
f(x + h) - f(x) = x + h - 5 - x + 5
\][/tex]
Combine like terms:
[tex]\[
f(x + h) - f(x) = h
\][/tex]
Now, substituting [tex]\( h = 2 \)[/tex] (since [tex]\( h \neq 0 \)[/tex]) to find numerical values:
1. [tex]\( f(h) = 2 - 5 = -3 \)[/tex].
2. [tex]\( f(x + h) = (x + 2) - 5 \Rightarrow f(x + 2) = x + 2 - 5 = x - 3 \Rightarrow \text{If we evaluate at} \, x = 2, \, f(2 + 2) = f(4) = 4 - 5 = -1 \)[/tex].
3. [tex]\( f(x + h) - f(x) = 2 \)[/tex] (as calculated directly from the general expression above).
So, the results are:
- [tex]\( f(h) = -3 \)[/tex]
- [tex]\( f(x+h) \)[/tex] when [tex]\( x = 2 \)[/tex] is [tex]\( -1 \)[/tex]
- [tex]\( f(x + h) - f(x) \)[/tex] is [tex]\( 2 \)[/tex]
Thus, we have found that [tex]\( f(h) = -3 \)[/tex], [tex]\( f(x + h) \)[/tex] evaluated at [tex]\( x = 2 \)[/tex] is [tex]\( -1 \)[/tex], and the difference [tex]\( f(x + h) - f(x) = 2 \)[/tex].
