Question

homework assignment 1.1: functions and function notation
score: 10.75/15 answered: 11/15
question 12
given the function $f(x) = 5x + 7$, evaluate and simplify the expressions below. see special instructions on how to enter your answers.
$f(a) = \square$
$f(a + h) = \square$
$\frac{f(a + h) - f(a)}{h} = \square$
instructions: simplify answers as much as possible. expressions such as $4(x + 2)$ and $(x + 5)^2$ should be expanded. also collect like terms, so $3x + x$ should be written as $4x$.
question help: video 1 video 2 written example 1

Explanation:

Step1: Evaluate \( f(a) \)

To find \( f(a) \), we substitute \( x = a \) into the function \( f(x) = 5x + 7 \).
\( f(a) = 5a + 7 \)

Step2: Evaluate \( f(a + h) \)

Substitute \( x = a + h \) into the function \( f(x) = 5x + 7 \).
\( f(a + h) = 5(a + h) + 7 \)
Expand the expression: \( 5a + 5h + 7 \)

Step3: Evaluate \( \frac{f(a + h) - f(a)}{h} \)

First, substitute the expressions we found for \( f(a + h) \) and \( f(a) \):
\( \frac{(5a + 5h + 7) - (5a + 7)}{h} \)
Simplify the numerator:
\( (5a + 5h + 7 - 5a - 7) = 5h \)
Then divide by \( h \):
\( \frac{5h}{h} = 5 \) (assuming \( h
eq 0 \))

Answer:

\( f(a) = \boldsymbol{5a + 7} \)
\( f(a + h) = \boldsymbol{5a + 5h + 7} \)
\( \frac{f(a + h) - f(a)}{h} = \boldsymbol{5} \)