Let's analyze each of these functions, sketch their graphs, and state their domains and ranges.

a) $f(x) = -(x - 4)(x + 2)(x - 1)$

Analysis:

• This is a cubic polynomial with three roots at $x = 4$, $x = -2$, and $x = 1$.
• The negative sign reflects the graph over the $x$-axis.
• As $x \to \infty$, $f(x) \to -\infty$, and as $x \to -\infty$, $f(x) \to \infty$.

Domain:

• The domain is all real numbers: $\mathbb{R}$ or $(-\infty, \infty)$.

Range:

• The range is also all real numbers: $\mathbb{R}$ or $(-\infty, \infty)$.

b) $g(x) = \frac{x + 2}{x - 1}$

Analysis:

• This is a rational function with a vertical asymptote at $x = 1$ (denominator equals zero).
• A horizontal asymptote exists at $y = 1$ because the degrees of the numerator and denominator are the same.

Domain:

• All real numbers except $x = 1$: $(-\infty, 1) \cup (1, \infty)$.

Range:

• All real numbers except $y = 1$: $(-\infty, 1) \cup (1, \infty)$.

c) $f : x \to -\log(x - 1)$

Analysis:

• This is a logarithmic function shifted right by 1 and reflected over the $x$-axis.
• The logarithm is defined when $x - 1 > 0$ or $x > 1$.
• As $x \to 1^+$, $f(x) \to -\infty$, and as $x \to \infty$, $f(x) \to \infty$.

Domain:

• $x > 1$ or $(1, \infty)$.

Range:

• All real numbers: $(-\infty, \infty)$.

d) $g : x \to 3 \cos x + 2$

Analysis:

• This is a cosine function scaled by 3 and shifted vertically by 2.
• The standard cosine function oscillates between $-1$ and $1$, so $3 \cos x$ oscillates between $-3$ and $3$.
• Adding 2 shifts the range to $[-1, 5]$.

Domain:

• All real numbers: $\mathbb{R}$ or $(-\infty, \infty)$.

Range:

• $[-1, 5]$.

Would you like more details on the graph sketches or explanations?

Related Questions

1. How do you find vertical and horizontal asymptotes of a rational function?
2. How does the reflection affect the graph of a cubic polynomial?
3. What is the general form of a logarithmic function and its properties?
4. How does scaling and shifting affect the range of trigonometric functions?
5. How do you determine the domain of composite functions?

Tip

When dealing with rational functions, always check where the denominator is zero to identify vertical asymptotes and restrictions on the domain.