Let's solve the problem step by step.
1. Understanding the [tex]$y$[/tex]-intercept of function [tex]\( m \)[/tex]:
- The [tex]$y$[/tex]-intercept of a function is the value of the function when [tex]\( x = 0 \)[/tex].
- Given the values in the table for function [tex]\( m \)[/tex]:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & -2 & 0 & 2 & 4 \\
\hline
m(x) & 4 & -6 & 0 & 70 \\
\hline
\end{array}
\][/tex]
- When [tex]\( x = 0 \)[/tex], the value of [tex]\( m(x) \)[/tex] is [tex]\( -6 \)[/tex].
- Therefore, the [tex]$y$[/tex]-intercept of [tex]\( m \)[/tex] is [tex]\(-6\)[/tex].
2. Understanding the [tex]$y$[/tex]-intercept of function [tex]\( n \)[/tex]:
- Function [tex]\( n \)[/tex] is a cubic function that passes through the points [tex]\((-1, 0)\)[/tex] and [tex]\((0, 2)\)[/tex].
- The [tex]$y$[/tex]-intercept of a function is the value of the function when [tex]\( x = 0 \)[/tex].
- Given the point [tex]\((0, 2)\)[/tex], when [tex]\( x = 0 \)[/tex], the value of [tex]\( n(x) \)[/tex] is [tex]\( 2 \)[/tex].
- Therefore, the [tex]$y$[/tex]-intercept of [tex]\( n \)[/tex] is [tex]\( 2 \)[/tex].
3. Comparing [tex]$y$[/tex]-intercepts:
- The [tex]$y$[/tex]-intercept of [tex]\( m \)[/tex] is [tex]\(-6\)[/tex].
- The [tex]$y$[/tex]-intercept of [tex]\( n \)[/tex] is [tex]\( 2 \)[/tex].
To find the correct statement among the given options:
- Option A: This is incorrect because the [tex]$y$[/tex]-intercept of [tex]\( m \)[/tex] is given as [tex]\( -6 \)[/tex].
- Option B: This is incorrect because the [tex]$y$[/tex]-intercept of [tex]\( m \)[/tex] is [tex]\(-6\)[/tex] and the [tex]$y$[/tex]-intercept of [tex]\( n \)[/tex] is [tex]\( 2 \)[/tex], so they are not equal.
- Option C: This is incorrect because [tex]\(-6\)[/tex] is not greater than [tex]\( 2 \)[/tex].
- Option D: This is correct because [tex]\(-6\)[/tex] is less than [tex]\( 2 \)[/tex].
Thus, the correct statement is:
D. The [tex]$y$[/tex]-intercept of [tex]\( m \)[/tex] is less than the [tex]$y$[/tex]-intercept of [tex]\( n \)[/tex].
