If possible, find the indicated values for the rational function. If the value does not exist, enter Does not exist. Find f(4), f(7), f(0). Then find domain of f. f(x)= \frac{3x^{2}+x+5}{x-4}

Given the rational function:

\[
f(x) = \frac{3x^2 + x + 5}{x - 4}
\]

### 1. Find \( f(4) \)

Substitute \( x = 4 \) into the function:

\[
f(4) = \frac{3(4)^2 + 4 + 5}{4 - 4} = \frac{3(16) + 4 + 5}{0} = \frac{48 + 4 + 5}{0} = \frac{57}{0}
\]

Since the denominator becomes zero, \( f(4) \) **does not exist**.

### 2. Find \( f(7) \)

Substitute \( x = 7 \) into the function:

\[
f(7) = \frac{3(7)^2 + 7 + 5}{7 - 4} = \frac{3(49) + 7 + 5}{3} = \frac{147 + 7 + 5}{3} = \frac{159}{3} = 53
\]

So, \( f(7) = 53 \).

### 3. Find \( f(0) \)

Substitute \( x = 0 \) into the function:

\[
f(0) = \frac{3(0)^2 + 0 + 5}{0 - 4} = \frac{0 + 0 + 5}{-4} = \frac{5}{-4} = -\frac{5}{4}
\]

So, \( f(0) = -\frac{5}{4} \).

### 4. Find the domain of \( f \)

The function \( f(x) \) is undefined when the denominator is zero. The denominator is \( x - 4 \), so the function is undefined when \( x = 4 \).

Thus, the domain of \( f \) is all real numbers except \( x = 4 \). In interval notation, the domain is:

\[
(-\infty, 4) \cup (4, \infty)
\]

### Summary of values:
- \( f(4) \): Does not exist
- \( f(7) = 53 \)
- \( f(0) = -\frac{5}{4} \)
- Domain: \( (-\infty, 4) \cup (4, \infty) \)

Would you like more details on any step?

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### Related Questions:
1. What happens to the function near \( x = 4 \)?
2. How does the function behave for large positive and negative values of \( x \)?
3. What are vertical and horizontal asymptotes of \( f(x) \)?
4. How would you find the intercepts of the function?
5. Can you calculate the inverse of this rational function?

### Tip:
Always check for discontinuities (like where the denominator is zero) when dealing with rational functions.

This detailed solution shows how to evaluate the rational function f(x) = (3x^2 + x + 5) / (x - 4) at different points and find its domain. Learn how to find f(4), f(7), f(0), and determine the domain of the function. Suitable for high school students in Grades 9-12.

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