Find the difference quotient and simplify your answer. $$ f(t)=\frac{1}{t-9}, \frac{f(t)-f(8)}{t-8}, \quad t \neq 8 $$

Question

UpStudy AI · Accepted Answer

To find the difference quotient for the function $$ f(t) = \frac{1}{t-9} $$, we need to compute $$ \frac{f(t) - f(8)}{t - 8} $$.

First, we calculate $$ f(8) $$:

$$
f(8) = \frac{1}{8 - 9} = \frac{1}{-1} = -1
$$

Now, we can substitute $$ f(t) $$ and $$ f(8) $$ into the difference quotient:

$$
\frac{f(t) - f(8)}{t - 8} = \frac{\frac{1}{t-9} - (-1)}{t - 8} = \frac{\frac{1}{t-9} + 1}{t - 8}
$$

Next, we simplify the numerator:

$$
\frac{1}{t-9} + 1 = \frac{1}{t-9} + \frac{t-9}{t-9} = \frac{1 + (t-9)}{t-9} = \frac{t - 8}{t - 9}
$$

Now, substituting this back into the difference quotient gives:

$$
\frac{\frac{t - 8}{t - 9}}{t - 8}
$$

We can simplify this expression. Since $$ t \neq 8 $$, we can cancel $$ t - 8 $$ in the numerator and denominator:

$$
\frac{t - 8}{t - 9} \cdot \frac{1}{t - 8} = \frac{1}{t - 9}
$$

Thus, the simplified form of the difference quotient is:

$$
\frac{1}{t - 9}
$$

Therefore, the final answer is:

$$
\frac{1}{t - 9}
$$
The simplified form of the difference quotient is $$ \frac{1}{t - 9} $$.

Find the difference quotient and simplify your answer.