Questions: f(x)=x-6 g(x)=3/x FIND f/g, THEN GIVE ITS DOMAIN using an interval OR UNION OF INTERVALS SIMPLIFY YOUR ANSWER

f(x)=x-6 g(x)=3/x

FIND f/g, THEN GIVE ITS DOMAIN using an interval OR UNION OF INTERVALS SIMPLIFY YOUR ANSWER

Transcript text: \[ f(x)=x-6 \quad g(x)=\frac{3}{x} \] FIND $\frac{f}{g}$, THEN GIVE ITS DOMAin using an interval OR UNION OF INTERVALS SIMPLIFY YOUR ANSWER

Solution

Solution Steps

To find \(\frac{f}{g}\), we need to divide the function \(f(x)\) by \(g(x)\). This involves substituting the expressions for \(f(x)\) and \(g(x)\) into the division. After simplifying the expression, we determine the domain by identifying the values of \(x\) that make the denominator zero, as these values are excluded from the domain.

Step 1: Find \(\frac{f}{g}\)

We start with the functions \(f(x) = x - 6\) and \(g(x) = \frac{3}{x}\). To find \(\frac{f}{g}\), we compute:

\[ \frac{f}{g} = \frac{x - 6}{\frac{3}{x}} = (x - 6) \cdot \frac{x}{3} = \frac{x(x - 6)}{3} \]

Step 2: Simplify the Expression

The simplified expression for \(\frac{f}{g}\) is:

\[ \frac{f}{g} = \frac{x^2 - 6x}{3} \]

Step 3: Determine the Domain

The domain of the function \(\frac{f}{g}\) is determined by the values of \(x\) that do not make the denominator zero. The denominator \(g(x) = \frac{3}{x}\) is undefined when \(x = 0\). Therefore, the domain excludes \(x = 0\).

Thus, the domain in interval notation is:

\[ (-\infty, 0) \cup (0, \infty) \]

Final Answer

The simplified function is \(\frac{f}{g} = \frac{x^2 - 6x}{3}\) and the domain is \((- \infty, 0) \cup (0, \infty)\).

\[ \boxed{\frac{f}{g} = \frac{x^2 - 6x}{3}, \text{ Domain: } (-\infty, 0) \cup (0, \infty)} \]

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A study examined whether the content of TV shows influenced the ability of viewers to recall brand names of items featured in the commercials. Analysts recorded the number of advertised brand names a group of people recalled immediately after watching TV shows and then that same group of people recalled 24 hours later. Strangely enough, it appears that they remembered more about the ads the next day. Should we conclude this is true in general about people's memory of TV ads? Complete parts a through d. c) How might the design of this experiment have tainted these results? A. Since the populations of each group are not normally distributed, subjects with high recall could skew the means for each group. B. Since the populations of each group are not normally distributed, a two-sample t-test is not appropriate to compare the means between the two groups. C. Since the subjects were not selected at random, the first interview might have helped the subjects with high recall to remember the brand names for a longer period of time than they would have otherwise. D. A subject with high recall right after the show might have high recall 24 hours later as well. Also, the first interview might have helped the subjects to remember the brand names for a longer period of time than they would have otherwise. d) Suggest a design that could compare immediate brand name recall with recall one day later. A. Randomly assign half of the group watching violent TV shows to be interviewed immediately after watching, while the other half of the group is interviewed 24 hours later. B. Assign the older half of the group watching violent TV shows to be interviewed immediately after watching, while the younger half of the group is interviewed 24 hours later. C. Assign subjects in the group with horrible short-term memory to be interviewed immediately after watching the violent TV show, and assign subjects with horrible long-term memory to be interviewed 24 hours later. D. Assign subjects in the group with good short-term memory to be interviewed immediately after watching the violent TV show, and assign subjects with good long-term memory to be interviewed 24 hours later.

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