Question

homework assignment 1.1: functions and function notation
score: 12.75/15 answered: 13/15
question 14
for the function $f(x) = 2x + 4$, evaluate and fully simplify each of the following.
$f(x + h) = $
$\frac{f(x + h) - f(x)}{h} = $

Explanation:

Step1: Substitute \( x + h \) into \( f(x) \)

To find \( f(x + h) \) for the function \( f(x) = 2x + 4 \), we replace every \( x \) in the function with \( x + h \). So we have:
\( f(x + h) = 2(x + h) + 4 \)

Step2: Distribute the 2

Using the distributive property \( a(b + c) = ab + ac \), we distribute the 2 in \( 2(x + h) \):
\( f(x + h) = 2x + 2h + 4 \)

Step3: Find \( f(x + h) - f(x) \)

First, we know \( f(x) = 2x + 4 \) and we just found \( f(x + h) = 2x + 2h + 4 \). Subtract \( f(x) \) from \( f(x + h) \):
\( f(x + h) - f(x) = (2x + 2h + 4) - (2x + 4) \)
Simplify the right - hand side:
\( f(x + h) - f(x)=2x + 2h + 4 - 2x - 4 = 2h \)

Step4: Divide by \( h \) ( \( h

eq0 \))
Now, we want to find \( \frac{f(x + h)-f(x)}{h} \). We know \( f(x + h)-f(x)=2h \), so:
\( \frac{f(x + h)-f(x)}{h}=\frac{2h}{h} \)
Since \( h
eq0 \), we can cancel out the \( h \) terms, and we get \( 2 \).

Answer:

s:
\( f(x + h)=\boldsymbol{2x + 2h + 4} \)

\( \frac{f(x + h)-f(x)}{h}=\boldsymbol{2} \)