To analyze the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] based on the given values, let's break down the steps:
1. Given Data:
- The table provides the values for [tex]\( g(x) \)[/tex] at specific points:
[tex]\[
\begin{array}{|c|c|}
\hline
x & g(x) \\
\hline
-5 & 4 \\
-4 & 1 \\
-3 & 0 \\
\hline
-2 & 1 \\
\hline
\end{array}
\][/tex]
2. Assumption about [tex]\( f(x) \)[/tex]:
- Let's assume [tex]\( f(x) \)[/tex] is given by the function [tex]\( f(x) = x + 1 \)[/tex].
3. Calculating [tex]\( f(x) \)[/tex]:
- For [tex]\( x = -5 \)[/tex]:
[tex]\[
f(-5) = -5 + 1 = -4
\][/tex]
- For [tex]\( x = -4 \)[/tex]:
[tex]\[
f(-4) = -4 + 1 = -3
\][/tex]
- For [tex]\( x = -3 \)[/tex]:
[tex]\[
f(-3) = -3 + 1 = -2
\][/tex]
- For [tex]\( x = -2 \)[/tex]:
[tex]\[
f(-2) = -2 + 1 = -1
\][/tex]
Thus, the values for [tex]\( f(x) \)[/tex] at these points are:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-5 & -4 \\
-4 & -3 \\
-3 & -2 \\
\hline
-2 & -1 \\
\hline
\end{array}
\][/tex]
4. Comparing [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
- At [tex]\( x = -5 \)[/tex]:
[tex]\[
f(-5) = -4, \quad g(-5) = 4
\][/tex]
- At [tex]\( x = -4 \)[/tex]:
[tex]\[
f(-4) = -3, \quad g(-4) = 1
\][/tex]
- At [tex]\( x = -3 \)[/tex]:
[tex]\[
f(-3) = -2, \quad g(-3) = 0
\][/tex]
- At [tex]\( x = -2 \)[/tex]:
[tex]\[
f(-2) = -1, \quad g(-2) = 1
\][/tex]
5. Checking Intersections:
- The functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] intersect where [tex]\( f(x) = g(x) \)[/tex].
- From the above comparisons, we note that [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] do not intersect at any of these points.
- Therefore, there are no points where they intersect.
6. Checking [tex]\( f(x) \)[/tex] Greater than [tex]\( g(x) \)[/tex]:
- To determine if [tex]\( f(x) \)[/tex] is greater than [tex]\( g(x) \)[/tex] for all given [tex]\( x \)[/tex]:
- At [tex]\( x = -5 \)[/tex]: [tex]\( f(-5) = -4 \)[/tex] and [tex]\( g(-5) = 4 \)[/tex]. Here, [tex]\( f(-5) < g(-5) \)[/tex].
- At [tex]\( x = -4 \)[/tex]: [tex]\( f(-4) = -3 \)[/tex] and [tex]\( g(-4) = 1 \)[/tex]. Here, [tex]\( f(-4) < g(-4) \)[/tex].
- At [tex]\( x = -3 \)[/tex]: [tex]\( f(-3) = -2 \)[/tex] and [tex]\( g(-3) = 0 \)[/tex]. Here, [tex]\( f(-3) < g(-3) \)[/tex].
- At [tex]\( x = -2 \)[/tex]: [tex]\( f(-2) = -1 \)[/tex] and [tex]\( g(-2) = 1 \)[/tex]. Here, [tex]\( f(-2) < g(-2) \)[/tex].
- Hence, [tex]\( f(x) \)[/tex] is not greater than [tex]\( g(x) \)[/tex] for any given [tex]\( x \)[/tex].
Conclusion:
- The statement "f(x) and g(x) intersect at exactly one point" is false.
- The statement "f(x) is greater than g(x) for all values of x" is also false.
Thus, neither statement about the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] is true.
