To analyze the function $f(x) = \frac{-x^2 + 2x - 1}{x - 1}$, let's go through each characteristic step-by-step.

1. Horizontal Asymptote

To find the horizontal asymptote of a rational function, compare the degrees of the polynomial in the numerator and the denominator:

• The numerator $-x^2 + 2x - 1$ is a degree 2 polynomial.
• The denominator $x - 1$ is a degree 1 polynomial.

Since the degree of the numerator (2) is greater than the degree of the denominator (1), there is no horizontal asymptote. Instead, there may be an oblique asymptote, which we can explore if needed.

2. Vertical Asymptote

A vertical asymptote occurs when the denominator equals zero, as long as the numerator does not also equal zero at that point.

• Setting $x - 1 = 0$, we find that $x = 1$.
• Substituting $x = 1$ in the numerator: $-1^2 + 2 \cdot 1 - 1 = -1 + 2 - 1 = 0$.

Since the numerator also equals zero when $x = 1$, there is no vertical asymptote at $x = 1$. Instead, there is a hole at $x = 1$.

3. x-Intercept

The x-intercepts occur where $f(x) = 0$, which means the numerator equals zero.

• Solving $-x^2 + 2x - 1 = 0$, we can factor this as: $-x^2 + 2x - 1 = -(x - 1)^2 = 0$ So, $x = 1$ is a root.

However, we previously found that $x = 1$ leads to both the numerator and the denominator being zero, which indicates a hole rather than an x-intercept. So, there is no x-intercept for this function.

4. y-Intercept

The y-intercept is found by setting $x = 0$.

• Substituting $x = 0$ in $f(x)$: $f(0) = \frac{-0^2 + 2 \cdot 0 - 1}{0 - 1} = \frac{-1}{-1} = 1$ Therefore, the y-intercept is $(0, 1)$.

5. Hole

As calculated earlier, a hole occurs at $x = 1$ because both the numerator and denominator equal zero at this point.

• To find the y-coordinate of the hole, simplify $f(x)$ around $x = 1$:
  • Factor out $(x - 1)$ from the numerator: $f(x) = \frac{-(x - 1)(x - 1)}{x - 1} = -(x - 1)$
  • Canceling $(x - 1)$, we get $f(x) = -(x - 1)$ for $x \neq 1$.
  • At $x = 1$, $f(x) \approx -(1 - 1) = 0$.

So, the hole is at $(1, 0)$.

Summary of Features

• Horizontal Asymptote: None
• Vertical Asymptote: None
• x-Intercept: None
• y-Intercept: $(0, 1)$
• Hole: $(1, 0)$

Would you like a further breakdown of any of these steps, or have any questions?

Related Questions

1. How do you determine if a function has an oblique asymptote?
2. What is the difference between a hole and a vertical asymptote in a rational function?
3. How do you find the x- and y-intercepts for other types of functions?
4. Why does a rational function with equal degrees in numerator and denominator have a horizontal asymptote?
5. Can a rational function have both a vertical asymptote and a hole at the same x-value?

Tip

For rational functions, always check for common factors in the numerator and denominator to identify holes.