Question

which of the following rational functions is graphed below? a. $f(x) = \frac{1}{(x - 2)(x - 4)}$ b. $f(x) = \frac{x}{(x - 2)(x - 4)}$ c. $f(x) = \frac{2}{(x - 2)(x - 4)}$ d. $f(x) = \frac{4}{(x - 2)(x - 4)}$

Explanation:

Step1: Analyze Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero (and numerator non - zero). From the graph, vertical asymptotes are at \(x = 2\) and \(x=4\) (assuming the x - values of the vertical asymptotes from the graph). So the denominator should be \((x - 2)(x - 4)\) (or \((x - 4)(x - 2)\)).

Step2: Analyze x - intercept

The x - intercept is where \(y = 0\), i.e., numerator is zero (denominator non - zero). From the graph, the x - intercept is at \(x = 0\) (since the graph crosses the x - axis at \(x = 0\)). So the numerator should be \(x\) (because when \(x=0\), the numerator is zero).

Now let's check the options:

• Option A: Numerator is \(1\), so x - intercept does not exist (since \(1

eq0\) for any \(x\)), so A is wrong.

• Option B: Numerator is \(x\) (x - intercept at \(x = 0\)) and denominator is \((x - 2)(x - 4)\) (vertical asymptotes at \(x = 2\) and \(x = 4\)), which matches the graph's features.
• Option C: The denominator seems to be miswritten, but assuming it's a typo, if we consider the structure, the numerator and denominator don't match the intercept and asymptote requirements.
• Option D: The numerator is \(1\), so x - intercept does not exist, so D is wrong.

Answer:

B. \(F(x)=\frac{x}{(x - 2)(x - 4)}\)