QuestionFind $f(3)$ given that $f^{\prime}(x) = \frac{x-8}{x-4}$ for $x \neq 4$ .

Studdy Solution

STEP 1

Assumptions1. The derivative of the function $f$ is given by $f^{\prime}: x \rightarrow \frac{x-8}{x-4}$ , where $x \neq4$ . . The original function $f(x)$ is not provided.
3. We need to find the value of $f(3)$ .

STEP 2

Since we have the derivative of the function, we can find the original function $f(x)$ by integrating $f^{\prime}(x)$ .
$f(x) = \int f^{\prime}(x) \, dx$

STEP 3

Now, plug in the given derivative function into the integral.
$f(x) = \int \frac{x-8}{x-} \, dx$

STEP 4

This integral can be solved by using the method of partial fractions. We can rewrite the integrand as a sum of simpler fractions.
$\frac{x-8}{x-4} = A + \frac{B}{x-4}$

STEP 5

By comparing coefficients, we can solve for $A$ and $B$ . This gives $A=1$ and $B=-4$ .

STEP 6

Now, we can rewrite the integral with these values.
$f(x) = \int (1 - \frac{4}{x-4}) \, dx$

STEP 7

This integral can now be easily solved.
$f(x) = x -4 \ln |x-4| + C$

STEP 8

We don't know the value of the constant $C$ , but we can find it if we have a point that the function $f(x)$ passes through. However, we don't have such a point, so we can't find the exact value of $f(3)$ .

STEP 9

Without additional information, we can't find the exact value of $f(3)$ . The best we can do is to express it in terms of the unknown constant $C$ .
$f(3) =3 -4 \ln |3-4| + C =3 +4 \ln + C =3 + C$ So, the value of $f(3)$ is $3 + C$ .

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