We are given the function:

$f(x) = \frac{5x^2 - 1}{x^2}$

We will evaluate $f(x)$ for each of the specified values.

---

a) $f(3)$

Substitute $x = 3$ into the function:

$f(3) = \frac{5(3)^2 - 1}{(3)^2} = \frac{5(9) - 1}{9} = \frac{45 - 1}{9} = \frac{44}{9}$

So, $f(3) = \frac{44}{9}$.

---

b) $f(0)$

Substitute $x = 0$ into the function:

$f(0) = \frac{5(0)^2 - 1}{(0)^2}$

However, division by 0 is undefined. Thus, $f(0)$ is undefined.

---

c) $f(-x)$

Substitute $x = -x$ into the function:

$f(-x) = \frac{5(-x)^2 - 1}{(-x)^2}$

Since $(-x)^2 = x^2$:

$f(-x) = \frac{5x^2 - 1}{x^2}$

So, $f(-x) = f(x) = \frac{5x^2 - 1}{x^2}$.

---

d) $f(1+x)$

Substitute $x = 1 + x$ into the function:

$f(1+x) = \frac{5(1 + x)^2 - 1}{(1 + x)^2}$

First, expand $(1 + x)^2$:

$(1 + x)^2 = 1 + 2x + x^2$

Now substitute this into the function:

$f(1+x) = \frac{5(1 + 2x + x^2) - 1}{1 + 2x + x^2}$

Simplify the numerator:

$5(1 + 2x + x^2) - 1 = 5 + 10x + 5x^2 - 1 = 4 + 10x + 5x^2$

Thus:

$f(1+x) = \frac{4 + 10x + 5x^2}{1 + 2x + x^2}$

---

Let me know if you need more details or clarifications!

Expanding Understanding:

1. What happens if $f(x)$ is evaluated at any other negative number?
2. How would the domain of $f(x)$ change if it were defined as $\frac{5x^2 - 1}{x}$?
3. What is the symmetry of $f(x)$? Is it even, odd, or neither?
4. How does the graph of $f(x)$ behave as $x \to \infty$?
5. How would you find the derivative of $f(x)$?

Tip:

When simplifying fractions, always check for common factors in the numerator and denominator to reduce the expression fully.