Question

1. f(x) = \frac{...}{x - 4}

vertical asymptote: x = 4
horizontal asymptote: y =...
x - intercept: (4, 0)?
y - intercept: (0, -4)?

domain: {x | x ≠ 4}
range: {y | y ≠ -1}?

1. f(x) = \frac{...}{...}

vertical asymptote:
horizontal asymptote:
...

Explanation:

Step1: Identify the function

The function given is \( f(x)=\frac{4}{x - 4} \) (assuming from the handwritten note and context of rational functions).

Step2: Vertical Asymptote

For a rational function \( \frac{N(x)}{D(x)} \), vertical asymptotes occur where \( D(x)=0 \) (and numerator is not zero there). Here, \( D(x)=x - 4 \), so set \( x - 4=0 \), which gives \( x = 4 \). So vertical asymptote is \( x = 4 \).

Step3: Horizontal Asymptote

For rational functions, if degree of numerator (\( n \)) and degree of denominator (\( m \)):

• If \( n

Step4: x - intercept

To find x - intercept, set \( f(x)=0 \), i.e., \( \frac{4}{x - 4}=0 \). But \( \frac{4}{x - 4}=0 \) has no solution (since numerator is 4, not zero), so there is no x - intercept? Wait, but the handwritten note has (4,0) which might be a mistake. Wait, maybe the function was miswritten. Wait, if the function was \( f(x)=\frac{4}{x - 4} \), then x - intercept: set \( y = 0 \), \( 0=\frac{4}{x - 4}\Rightarrow 4 = 0 \), which is impossible. So no x - intercept. But the handwritten note has (4,0), which is on the vertical asymptote, so that's an error.

Step5: y - intercept

To find y - intercept, set \( x = 0 \), then \( f(0)=\frac{4}{0 - 4}=\frac{4}{-4}=- 1 \). Wait, the handwritten note has (0, - 4)? Wait, maybe the function is \( f(x)=\frac{4}{x - 1} \)? No, the vertical asymptote was written as \( x = 4 \). Wait, maybe the original function is \( f(x)=\frac{4}{x - 4} \), then y - intercept: \( x = 0 \), \( f(0)=\frac{4}{-4}=-1 \). But the handwritten note has (0, - 4), maybe a miscalculation.

Step6: Domain

Domain of a rational function is all real numbers except where denominator is zero. So \( x
eq4 \), so domain is \( \{x\in\mathbb{R}|x
eq4\} \).

Step7: Range

For \( y=\frac{4}{x - 4} \), solve for x: \( y(x - 4)=4\Rightarrow yx-4y = 4\Rightarrow yx=4 + 4y\Rightarrow x=\frac{4 + 4y}{y} \) (for \( y
eq0 \)). So y cannot be 0, so range is \( \{y\in\mathbb{R}|y
eq0\} \).

Answer:

For the function \( f(x)=\frac{4}{x - 4} \):

• Vertical Asymptote: \( x = 4 \)
• Horizontal Asymptote: \( y = 0 \)
• x - intercept: None (since \( \frac{4}{x - 4}=0 \) has no solution)
• y - intercept: \( (0,-1) \) (if calculated correctly, handwritten (0, - 4) is incorrect)
• Domain: \( \{x|x

eq4,x\in\mathbb{R}\} \)

• Range: \( \{y|y

eq0,y\in\mathbb{R}\} \)