Question

1. - / 5 points find the difference quotient and simplify your answer. ( f(t) = \frac{1}{t - 4}, \frac{f(t) - f(3)}{t - 3}, t

eq 4 )

Explanation:

Step1: Find \( f(3) \)

Substitute \( t = 3 \) into \( f(t)=\frac{1}{t - 4} \).
\( f(3)=\frac{1}{3 - 4}=\frac{1}{-1}=-1 \)

Step2: Substitute \( f(t) \) and \( f(3) \) into the difference quotient

The difference quotient is \( \frac{f(t)-f(3)}{t - 3} \), substitute \( f(t)=\frac{1}{t - 4} \) and \( f(3)=-1 \).
\( \frac{\frac{1}{t - 4}-(-1)}{t - 3}=\frac{\frac{1}{t - 4}+1}{t - 3} \)

Step3: Simplify the numerator

Find a common denominator for the numerator: \( \frac{1+(t - 4)}{t - 4}=\frac{1 + t-4}{t - 4}=\frac{t - 3}{t - 4} \)

Step4: Divide by \( t - 3 \)

Now the expression becomes \( \frac{\frac{t - 3}{t - 4}}{t - 3} \), since \( t
eq3 \), we can cancel \( t - 3 \) ( \( t - 3
eq0 \) ).
\( \frac{t - 3}{t - 4}\times\frac{1}{t - 3}=\frac{1}{t - 4} \)

Answer:

\( \frac{1}{t - 4} \)