To find the domain and range of the function $f(x) = \frac{x + 3}{5x - 5}$, let's proceed step by step.

1. Domain:

The domain of a function includes all possible values of $x$ for which the function is defined. The only restriction in this function comes from the denominator, which cannot be zero. So, we need to find the values of $x$ that make the denominator zero.

Set the denominator equal to zero and solve for $x$: $5x - 5 = 0$ $5x = 5$ $x = 1$

Thus, the function is undefined at $x = 1$. Therefore, the domain of $f(x)$ is all real numbers except $x = 1$. In interval notation, the domain is: $\text{Domain} = (-\infty, 1) \cup (1, \infty)$

2. Range:

To find the range of the function, consider the behavior of $f(x)$ as $x$ approaches different values, including the vertical asymptote at $x = 1$.

Rewriting the function in a different form might help us see the range more clearly: $f(x) = \frac{x + 3}{5(x - 1)} = \frac{1}{5} + \frac{2}{5(x - 1)}$

As $x$ approaches 1 from the left or right, $f(x)$ approaches infinity or negative infinity, showing that $f(x)$ can take on extremely large positive or negative values.

To further solidify the range, consider the behavior of the function as $x$ approaches infinity: $\lim_{x \to \infty} f(x) = \frac{x + 3}{5x - 5} \approx \frac{x}{5x} = \frac{1}{5}$

The function never actually reaches $y = \frac{1}{5}$ but can get arbitrarily close to it. Since the function can take on any real value except for the horizontal asymptote, the range of $f(x)$ is: $\text{Range} = (-\infty, \frac{1}{5}) \cup (\frac{1}{5}, \infty)$

Summary:

• Domain: $(-\infty, 1) \cup (1, \infty)$
• Range: $(-\infty, \frac{1}{5}) \cup (\frac{1}{5}, \infty)$

Would you like more details, or do you have any questions?

Here are 5 related questions to deepen your understanding:

1. How do you determine if a function has a horizontal asymptote?
2. What is the significance of the vertical asymptote in determining the domain?
3. How would the domain and range change if the function were $f(x) = \frac{x + 3}{5x + 5}$?
4. What are some other functions where the domain excludes a specific value of $x$?
5. How can you graphically determine the domain and range of a rational function?

Tip: When analyzing rational functions, always check where the denominator equals zero to find the domain and identify horizontal asymptotes by comparing the degrees of the numerator and denominator.