Questions: Consider the function f(x)=10/x. a. Complete the table of values below. x f(x)=10/x -1 -0.1 -0.001 0.001 0.1 1 b. Based on your work in part (a), which of the following are true? Select all that apply. As x → 0+, f(x) decreases without bound. As x → 0-, f(x) decreases without bound. As x → 0-, f(x) increases without bound. As x → 0+, f(x) increases without bound.

Consider the function f(x)=10/x.
a. Complete the table of values below.
x f(x)=10/x
-1
-0.1
-0.001
0.001
0.1
1
b. Based on your work in part (a), which of the following are true? Select all that apply.
As x → 0+, f(x) decreases without bound.
As x → 0-, f(x) decreases without bound.
As x → 0-, f(x) increases without bound.
As x → 0+, f(x) increases without bound.

Transcript text: imathas.rationalreasoning.net/assessment/showtest.php Unlimited attempts. Score on last attempt: 1. Score in gradebook: 1 Consider the function $f(x)=\frac{10}{x}$. a. Complete the table of values below. \begin{tabular}{|c|c|} \hline$x$ & $f(x)=\frac{10}{x}$ \\ \hline-1 & $\square$ \\ \hline-0.1 & $\square$ \\ \hline-0.001 & $\square$ \\ \hline 0.001 & $\square$ \\ \hline 0.1 & $\square$ \\ \hline 1 & $\square$ \\ \hline \end{tabular} b. Based on your work in part (a), which of the following are true? Select all that apply. As $x \rightarrow 0^{+}, f(x)$ decreases without bound. As $x \rightarrow 0^{-}, f(x)$ decreases without bound. As $x \rightarrow 0^{-}, f(x)$ increases without bound. As $x \rightarrow 0^{+}, f(x)$ increases without bound. Submit Question 3. Points possible: 2 Unlimited attempts.

Solution

Solution Steps

Solution Approach

a. To complete the table of values for the function $ f(x) = \frac{10}{x} $, substitute each given $ x $ value into the function to calculate $ f(x) $.

b. Analyze the behavior of $ f(x) $ as $ x $ approaches 0 from the positive side ($ x \rightarrow 0^{+} $) and from the negative side ($ x \rightarrow 0^{-} $). Determine whether $ f(x) $ increases or decreases without bound in each case.

Step 1: Calculate $ f(x) $ for Given Values

We evaluate the function $ f(x) = \frac{10}{x} $ for the specified $ x $ values:

For $ x = -1 $: \[ f(-1) = \frac{10}{-1} = -10.0 \]
For $ x = -0.1 $: \[ f(-0.1) = \frac{10}{-0.1} = -100.0 \]
For $ x = -0.001 $: \[ f(-0.001) = \frac{10}{-0.001} = -10000.0 \]
For $ x = 0.001 $: \[ f(0.001) = \frac{10}{0.001} = 10000.0 \]
For $ x = 0.1 $: \[ f(0.1) = \frac{10}{0.1} = 100.0 \]
For $ x = 1 $: \[ f(1) = \frac{10}{1} = 10.0 \]

The completed table of values is: \[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -1 & -10.0 \\ -0.1 & -100.0 \\ -0.001 & -10000.0 \\ 0.001 & 10000.0 \\ 0.1 & 100.0 \\ 1 & 10.0 \\ \hline \end{array} \]

Step 2: Analyze the Behavior of $ f(x) $ as $ x $ Approaches 0

We analyze the limits of $ f(x) $ as $ x $ approaches 0 from both the positive and negative sides:

As $ x \rightarrow 0^{+} $: \[ f(x) \rightarrow +\infty \quad \text{(increases without bound)} \]
As $ x \rightarrow 0^{-} $: \[ f(x) \rightarrow -\infty \quad \text{(decreases without bound)} \]