Answer :
To solve this problem, let's break it down step-by-step:
### Step 1: Identify the x-intercepts of Function A
To find the x-intercepts of Function A, we need to check where the function values, [tex]\(f(x)\)[/tex], are equal to 0. From the given data:
[tex]\[ \begin{tabular}{|c|c|} \hline$x$ & $f(x)$ \\ \hline -2 & 2 \\ \hline 0 & 1 \\ \hline 2 & 0 \\ \hline 4 & -1 \\ \hline 6 & 0 \\ \hline \end{tabular} \][/tex]
We see that the function values are zero at [tex]\(x = 2\)[/tex] and [tex]\(x = 6\)[/tex]. Therefore, the x-intercepts of Function A are [tex]\(2\)[/tex] and [tex]\(6\)[/tex].
### Step 2: Restate the Problem for Function B
We are given that:
- The x-intercept in Function B is one-third as large as the x-intercept in Function A.
### Step 3: Calculate the x-intercept for Function B
Since Function A has two x-intercepts, let's choose the first one for our calculations. We take [tex]\(x = 2\)[/tex] as the x-intercept of Function A.
To find the corresponding x-intercept in Function B:
[tex]\[ \text{x-intercept of Function B} = \frac{x \text{-intercept of Function A}}{3} \][/tex]
Using [tex]\(x = 2\)[/tex] from Function A:
[tex]\[ x_B = \frac{2}{3} \, \approx \, 0.67 \][/tex]
So the x-intercept of Function B is approximately [tex]\(0.67\)[/tex].
### Step 4: Analyze the Relationship Between x-intercepts
Now we address the given options:
1. The [tex]$x$[/tex]-intercept in function [tex]$B$[/tex] is one-third as large as the [tex]$x$[/tex]-intercept in function [tex]$A$[/tex].
- This is correct. We've calculated that the x-intercept of Function B, which is [tex]\(\approx 0.67\)[/tex], is one-third of the x-intercept of Function A, which is 2.
All other options can be disregarded as they don't match our calculated results and the given relationship.
### Conclusion
So the only correct conclusion is:
The [tex]$x$[/tex]-intercept in function [tex]$B$[/tex] is one-third as large as the [tex]$x$[/tex]-intercept in function [tex]$A$[/tex].
### Step 1: Identify the x-intercepts of Function A
To find the x-intercepts of Function A, we need to check where the function values, [tex]\(f(x)\)[/tex], are equal to 0. From the given data:
[tex]\[ \begin{tabular}{|c|c|} \hline$x$ & $f(x)$ \\ \hline -2 & 2 \\ \hline 0 & 1 \\ \hline 2 & 0 \\ \hline 4 & -1 \\ \hline 6 & 0 \\ \hline \end{tabular} \][/tex]
We see that the function values are zero at [tex]\(x = 2\)[/tex] and [tex]\(x = 6\)[/tex]. Therefore, the x-intercepts of Function A are [tex]\(2\)[/tex] and [tex]\(6\)[/tex].
### Step 2: Restate the Problem for Function B
We are given that:
- The x-intercept in Function B is one-third as large as the x-intercept in Function A.
### Step 3: Calculate the x-intercept for Function B
Since Function A has two x-intercepts, let's choose the first one for our calculations. We take [tex]\(x = 2\)[/tex] as the x-intercept of Function A.
To find the corresponding x-intercept in Function B:
[tex]\[ \text{x-intercept of Function B} = \frac{x \text{-intercept of Function A}}{3} \][/tex]
Using [tex]\(x = 2\)[/tex] from Function A:
[tex]\[ x_B = \frac{2}{3} \, \approx \, 0.67 \][/tex]
So the x-intercept of Function B is approximately [tex]\(0.67\)[/tex].
### Step 4: Analyze the Relationship Between x-intercepts
Now we address the given options:
1. The [tex]$x$[/tex]-intercept in function [tex]$B$[/tex] is one-third as large as the [tex]$x$[/tex]-intercept in function [tex]$A$[/tex].
- This is correct. We've calculated that the x-intercept of Function B, which is [tex]\(\approx 0.67\)[/tex], is one-third of the x-intercept of Function A, which is 2.
All other options can be disregarded as they don't match our calculated results and the given relationship.
### Conclusion
So the only correct conclusion is:
The [tex]$x$[/tex]-intercept in function [tex]$B$[/tex] is one-third as large as the [tex]$x$[/tex]-intercept in function [tex]$A$[/tex].
