Question

1 exploration: identifying graphs of rational functions (continued)
c.
d.
e.
f.
communicate your answer

1. what are some of the characteristics of the graph of a rational function?
2. determine the intercepts, asymptotes, domain, and range of the rational function

$g(x) = \frac{x - a}{x - b}$.

Explanation:

To determine the characteristics of a rational function's graph, we analyze key features:

1. Asymptotes: Vertical asymptotes occur where the denominator is zero (if the numerator isn't also zero there). Horizontal asymptotes depend on the degrees of the numerator and denominator. For \( g(x)=\frac{x - a}{x - b} \), the horizontal asymptote is \( y = 1 \) (since degrees of numerator and denominator are equal, the ratio of leading coefficients is \( 1/1 = 1 \)). Vertical asymptote is \( x = b \) (denominator zero, numerator at \( x = b \) is \( b - a \), which is non - zero unless \( a = b \), but if \( a = b \), it's a different function).
2. Intercepts:

• x - intercept: Set \( y = 0 \), so \( \frac{x - a}{x - b}=0 \). This implies \( x - a = 0 \) (since denominator can't be zero), so \( x = a \). Thus, the x - intercept is \( (a,0) \).
• y - intercept: Set \( x = 0 \), then \( y=\frac{0 - a}{0 - b}=\frac{-a}{-b}=\frac{a}{b} \) (assuming \( b

eq0 \)). So the y - intercept is \( (0,\frac{a}{b}) \).

1. Domain: The domain is all real numbers except \( x = b \) (since the denominator \( x - b

eq0 \)), so \( \text{Domain}=\{x\in\mathbb{R}|x
eq b\} \).

1. Range: To find the range, we can solve \( y=\frac{x - a}{x - b} \) for \( x \). Cross - multiply: \( y(x - b)=x - a \), \( yx - by=x - a \), \( yx - x=by - a \), \( x(y - 1)=by - a \), \( x=\frac{by - a}{y - 1} \). The denominator \( y - 1

eq0 \), so \( y
eq1 \). Thus, the range is \( \{y\in\mathbb{R}|y
eq1\} \).

Answer:

• Intercepts:
• x - intercept: \( (a,0) \)
• y - intercept: \( (0,\frac{a}{b}) \) (for \( b

eq0 \))

• Asymptotes:
• Vertical Asymptote: \( x = b \)
• Horizontal Asymptote: \( y = 1 \)
• Domain: \( \{x\in\mathbb{R}|x

eq b\} \)

• Range: \( \{y\in\mathbb{R}|y

eq1\} \)