Question

example 3:
$f(x)=-dfrac{2}{2x - 4}+1$
vertical asymptote:
horizontal asymptote:

domain:
range:

example 4:
$f(x)=dfrac{1}{x}-3$
vertical asymptote:
horizontal asymptote:

domain:
range:

Explanation:

Example 3: \( f(x) = -\frac{2}{2x - 4} + 1 \)

Vertical Asymptote:

Step 1: Find where denominator is zero

Set \( 2x - 4 = 0 \).
\( 2x = 4 \)

Step 2: Solve for x

\( x = \frac{4}{2} = 2 \)

Step 1: Analyze degree of numerator and denominator

For rational functions, if degree of numerator (here, numerator of the fractional part is constant, degree 0) is less than degree of denominator (degree 1 for \( 2x - 4 \)), the horizontal asymptote is \( y = \) the constant term added (after simplifying the fractional part's horizontal asymptote, which is \( y = 0 \) for \( \frac{\text{constant}}{\text{linear}} \), then we add 1).

Step 2: Determine horizontal asymptote

The horizontal asymptote of \( -\frac{2}{2x - 4} \) is \( y = 0 \), then adding 1 gives \( y = 1 \).

Step 1: Identify excluded value

From vertical asymptote, \( x = 2 \) is excluded.

Step 2: Write domain

All real numbers except \( x = 2 \), so \( (-\infty, 2) \cup (2, \infty) \)

Answer:

\( x = 2 \)

Horizontal Asymptote: