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Let's analyze the functions [tex]\( f(x) \)[/tex], [tex]\( g(x) \)[/tex], and [tex]\( h(x) \)[/tex] based on the given table:
[tex]\[ \begin{array}{|c|c|c|c|} \hline x & f(x) & g(x) & h(x) \\ \hline -2 & -14 & \frac{1}{49} & -28 \\ \hline -1 & -7 & \frac{1}{7} & -7 \\ \hline 0 & 0 & 1 & 0 \\ \hline 1 & 7 & 7 & -7 \\ \hline 2 & 14 & 49 & -28 \\ \hline \end{array} \][/tex]
1. Y-Intercepts:
The [tex]\( y \)[/tex]-intercept of a function is the value of the function when [tex]\( x = 0 \)[/tex].
For [tex]\( f(x) \)[/tex]:
[tex]\[ f(0) = 0 \][/tex]
For [tex]\( g(x) \)[/tex]:
[tex]\[ g(0) = 1 \][/tex]
For [tex]\( h(x) \)[/tex]:
[tex]\[ h(0) = 0 \][/tex]
Therefore, only [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex] have [tex]\( y \)[/tex]-intercepts at [tex]\( y = 0 \)[/tex].
2. X-Intercepts:
The [tex]\( x \)[/tex]-intercept of a function is the value of [tex]\( x \)[/tex] for which the function [tex]\( f(x) = 0 \)[/tex].
For [tex]\( f(x) \)[/tex]:
When [tex]\( x = 0 \)[/tex], [tex]\( f(x) = 0 \)[/tex].
For [tex]\( g(x) \)[/tex]:
There is no [tex]\( x \)[/tex] for which [tex]\( g(x) = 0 \)[/tex] in the given table.
For [tex]\( h(x) \)[/tex]:
When [tex]\( x = 0 \)[/tex], [tex]\( h(x) = 0 \)[/tex].
Therefore, only [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex] have [tex]\( x \)[/tex]-intercepts.
3. Minimum Values:
[tex]\[ \text{Minimum of } f(x) = -14 \][/tex]
[tex]\[ \text{Minimum of } g(x) = \frac{1}{49} \approx 0.0204 \][/tex]
[tex]\[ \text{Minimum of } h(x) = -28 \][/tex]
The minimum of [tex]\( h(x) \)[/tex] is indeed less than the other minimum values.
4. Range of Values:
Let's find the unique values each function takes (i.e., the range).
For [tex]\( f(x) \)[/tex]:
[tex]\[ \{-14, -7, 0, 7, 14\} \][/tex]
For [tex]\( g(x) \)[/tex]:
[tex]\[ \left\{ \frac{1}{49}, \frac{1}{7}, 1, 7, 49 \right\} \][/tex]
For [tex]\( h(x) \)[/tex]:
[tex]\[ \{-28, -7, 0\} \][/tex]
Comparing the sizes of the sets:
[tex]\[ \text{Number of values in range of } f(x) = 5 \][/tex]
[tex]\[ \text{Number of values in range of } g(x) = 5 \][/tex]
[tex]\[ \text{Number of values in range of } h(x) = 3 \][/tex]
Therefore, the range of [tex]\( h(x) \)[/tex] does not have more values than the other ranges.
5. Maximum Values:
[tex]\[ \text{Maximum of } f(x) = 14 \][/tex]
[tex]\[ \text{Maximum of } g(x) = 49 \][/tex]
[tex]\[ \text{Maximum of } h(x) = 0 \][/tex]
The maximum of [tex]\( g(x) \)[/tex] is indeed greater than the other maximum values.
Based on this analysis, the following statements are true:
1. Only [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex] have [tex]\( y \)[/tex]-intercepts.
2. Only [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex] have [tex]\( x \)[/tex]-intercepts.
3. The minimum of [tex]\( h(x) \)[/tex] is less than the other minimums.
4. The maximum of [tex]\( g(x) \)[/tex] is greater than the other maximums.
However, the statement about [tex]\( h(x) \)[/tex] having more values in its range is not true.
Thus, the three correct options are:
- Only [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex] have [tex]\( y \)[/tex]-intercepts.
- Only [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex] have [tex]\( x \)[/tex]-intercepts.
- The minimum of [tex]\( h(x) \)[/tex] is less than the other minimums.
[tex]\[ \begin{array}{|c|c|c|c|} \hline x & f(x) & g(x) & h(x) \\ \hline -2 & -14 & \frac{1}{49} & -28 \\ \hline -1 & -7 & \frac{1}{7} & -7 \\ \hline 0 & 0 & 1 & 0 \\ \hline 1 & 7 & 7 & -7 \\ \hline 2 & 14 & 49 & -28 \\ \hline \end{array} \][/tex]
1. Y-Intercepts:
The [tex]\( y \)[/tex]-intercept of a function is the value of the function when [tex]\( x = 0 \)[/tex].
For [tex]\( f(x) \)[/tex]:
[tex]\[ f(0) = 0 \][/tex]
For [tex]\( g(x) \)[/tex]:
[tex]\[ g(0) = 1 \][/tex]
For [tex]\( h(x) \)[/tex]:
[tex]\[ h(0) = 0 \][/tex]
Therefore, only [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex] have [tex]\( y \)[/tex]-intercepts at [tex]\( y = 0 \)[/tex].
2. X-Intercepts:
The [tex]\( x \)[/tex]-intercept of a function is the value of [tex]\( x \)[/tex] for which the function [tex]\( f(x) = 0 \)[/tex].
For [tex]\( f(x) \)[/tex]:
When [tex]\( x = 0 \)[/tex], [tex]\( f(x) = 0 \)[/tex].
For [tex]\( g(x) \)[/tex]:
There is no [tex]\( x \)[/tex] for which [tex]\( g(x) = 0 \)[/tex] in the given table.
For [tex]\( h(x) \)[/tex]:
When [tex]\( x = 0 \)[/tex], [tex]\( h(x) = 0 \)[/tex].
Therefore, only [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex] have [tex]\( x \)[/tex]-intercepts.
3. Minimum Values:
[tex]\[ \text{Minimum of } f(x) = -14 \][/tex]
[tex]\[ \text{Minimum of } g(x) = \frac{1}{49} \approx 0.0204 \][/tex]
[tex]\[ \text{Minimum of } h(x) = -28 \][/tex]
The minimum of [tex]\( h(x) \)[/tex] is indeed less than the other minimum values.
4. Range of Values:
Let's find the unique values each function takes (i.e., the range).
For [tex]\( f(x) \)[/tex]:
[tex]\[ \{-14, -7, 0, 7, 14\} \][/tex]
For [tex]\( g(x) \)[/tex]:
[tex]\[ \left\{ \frac{1}{49}, \frac{1}{7}, 1, 7, 49 \right\} \][/tex]
For [tex]\( h(x) \)[/tex]:
[tex]\[ \{-28, -7, 0\} \][/tex]
Comparing the sizes of the sets:
[tex]\[ \text{Number of values in range of } f(x) = 5 \][/tex]
[tex]\[ \text{Number of values in range of } g(x) = 5 \][/tex]
[tex]\[ \text{Number of values in range of } h(x) = 3 \][/tex]
Therefore, the range of [tex]\( h(x) \)[/tex] does not have more values than the other ranges.
5. Maximum Values:
[tex]\[ \text{Maximum of } f(x) = 14 \][/tex]
[tex]\[ \text{Maximum of } g(x) = 49 \][/tex]
[tex]\[ \text{Maximum of } h(x) = 0 \][/tex]
The maximum of [tex]\( g(x) \)[/tex] is indeed greater than the other maximum values.
Based on this analysis, the following statements are true:
1. Only [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex] have [tex]\( y \)[/tex]-intercepts.
2. Only [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex] have [tex]\( x \)[/tex]-intercepts.
3. The minimum of [tex]\( h(x) \)[/tex] is less than the other minimums.
4. The maximum of [tex]\( g(x) \)[/tex] is greater than the other maximums.
However, the statement about [tex]\( h(x) \)[/tex] having more values in its range is not true.
Thus, the three correct options are:
- Only [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex] have [tex]\( y \)[/tex]-intercepts.
- Only [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex] have [tex]\( x \)[/tex]-intercepts.
- The minimum of [tex]\( h(x) \)[/tex] is less than the other minimums.
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