Answer :
Let's analyze step-by-step the characteristics of the functions [tex]\( f(x) \)[/tex], [tex]\( g(x) \)[/tex], and [tex]\( h(x) \)[/tex] given in the table.
### Step 1: Y-Intercepts
The [tex]\( y \)[/tex]-intercept 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, both [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex] have [tex]\( y \)[/tex]-intercepts at 0, while [tex]\( g(x) \)[/tex] does not have a [tex]\( y \)[/tex]-intercept of 0.
Statement: Only [tex]\( f(x) \)[/tex] and [tex]\( h(x)\)[/tex] have y-intercepts. This is True.
### Step 2: X-Intercepts
The [tex]\( x \)[/tex]-intercept is the value of [tex]\( x \)[/tex] when the function value is 0.
- For [tex]\( f(x) \)[/tex]: [tex]\( f(x) = 0 \)[/tex] when [tex]\( x = 0 \)[/tex].
- For [tex]\( g(x) \)[/tex]: [tex]\( g(x) \)[/tex] does not equal 0 for any given [tex]\( x \)[/tex] in the table.
- For [tex]\( h(x) \)[/tex]: [tex]\( h(x) = 0 \)[/tex] when [tex]\( x = 0 \)[/tex].
Therefore, both [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex] have [tex]\( x \)[/tex]-intercepts at 0, whereas [tex]\( g(x) \)[/tex] does not cross the [tex]\( x \)[/tex]-axis.
Statement: Only [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex] have x-intercepts. This is True.
### Step 3: Minimum Values
We compare the minimum values of each function from the given data.
- Minimum of [tex]\( f(x) \)[/tex]: [tex]\(\text{min}(f(x)) = \text{min}(-14, -7, 0, 7, 14) = -14\)[/tex].
- Minimum of [tex]\( g(x) \)[/tex]: [tex]\(\text{min}(g(x)) = \text{min}(\frac{1}{49}, \frac{1}{7}, 1, 7, 49) = \frac{1}{49}\)[/tex].
- Minimum of [tex]\( h(x) \)[/tex]: [tex]\(\text{min}(h(x)) = \text{min}(-28, -7, 0, -7, -28) = -28\)[/tex].
Statement: The minimum of [tex]\( h(x) \)[/tex] is less than the other minimums. This is True.
### Step 4: Ranges
We determine the range of each function by looking at the set of values each function takes on.
- Range of [tex]\( f(x) \)[/tex]: [tex]\(\{-14, -7, 0, 7, 14\}\)[/tex].
- Range of [tex]\( g(x) \)[/tex]: [tex]\(\{\frac{1}{49}, \frac{1}{7}, 1, 7, 49\}\)[/tex].
- Range of [tex]\( h(x) \)[/tex]: [tex]\(\{-28, -7, 0, -7, -28\} \equiv \{-28, -7, 0\}\)[/tex].
- The range of [tex]\( f(x) \)[/tex] has 5 values.
- The range of [tex]\( g(x) \)[/tex] has 5 values.
- The range of [tex]\( h(x) \)[/tex] has 3 values.
Statement: The range of [tex]\( h(x) \)[/tex] has more values than the other ranges. This is False.
### Step 5: Maximum Values
We compare the maximum values of each function from the given data.
- Maximum of [tex]\( f(x) \)[/tex]: [tex]\(\text{max}(f(x)) = \text{max}(-14, -7, 0, 7, 14) = 14\)[/tex].
- Maximum of [tex]\( g(x) \)[/tex]: [tex]\(\text{max}(g(x)) = \text{max}(\frac{1}{49}, \frac{1}{7}, 1, 7, 49) = 49\)[/tex].
- Maximum of [tex]\( h(x) \)[/tex]: [tex]\(\text{max}(h(x)) = \text{max}(-28, -7, 0, -7, -28) = 0\)[/tex].
Statement: The maximum of [tex]\( g(x) \)[/tex] is greater than the other maximums. This is True.
### Final Answers
Based on the analysis, the true statements are:
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 range of [tex]\( h(x) \)[/tex] has more values than the other ranges. (False)
5. The maximum of [tex]\( g(x) \)[/tex] is greater than the other maximums.
Therefore, select these three options:
- 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.
- The maximum of [tex]\( g(x) \)[/tex] is greater than the other maximums.
### Step 1: Y-Intercepts
The [tex]\( y \)[/tex]-intercept 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, both [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex] have [tex]\( y \)[/tex]-intercepts at 0, while [tex]\( g(x) \)[/tex] does not have a [tex]\( y \)[/tex]-intercept of 0.
Statement: Only [tex]\( f(x) \)[/tex] and [tex]\( h(x)\)[/tex] have y-intercepts. This is True.
### Step 2: X-Intercepts
The [tex]\( x \)[/tex]-intercept is the value of [tex]\( x \)[/tex] when the function value is 0.
- For [tex]\( f(x) \)[/tex]: [tex]\( f(x) = 0 \)[/tex] when [tex]\( x = 0 \)[/tex].
- For [tex]\( g(x) \)[/tex]: [tex]\( g(x) \)[/tex] does not equal 0 for any given [tex]\( x \)[/tex] in the table.
- For [tex]\( h(x) \)[/tex]: [tex]\( h(x) = 0 \)[/tex] when [tex]\( x = 0 \)[/tex].
Therefore, both [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex] have [tex]\( x \)[/tex]-intercepts at 0, whereas [tex]\( g(x) \)[/tex] does not cross the [tex]\( x \)[/tex]-axis.
Statement: Only [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex] have x-intercepts. This is True.
### Step 3: Minimum Values
We compare the minimum values of each function from the given data.
- Minimum of [tex]\( f(x) \)[/tex]: [tex]\(\text{min}(f(x)) = \text{min}(-14, -7, 0, 7, 14) = -14\)[/tex].
- Minimum of [tex]\( g(x) \)[/tex]: [tex]\(\text{min}(g(x)) = \text{min}(\frac{1}{49}, \frac{1}{7}, 1, 7, 49) = \frac{1}{49}\)[/tex].
- Minimum of [tex]\( h(x) \)[/tex]: [tex]\(\text{min}(h(x)) = \text{min}(-28, -7, 0, -7, -28) = -28\)[/tex].
Statement: The minimum of [tex]\( h(x) \)[/tex] is less than the other minimums. This is True.
### Step 4: Ranges
We determine the range of each function by looking at the set of values each function takes on.
- Range of [tex]\( f(x) \)[/tex]: [tex]\(\{-14, -7, 0, 7, 14\}\)[/tex].
- Range of [tex]\( g(x) \)[/tex]: [tex]\(\{\frac{1}{49}, \frac{1}{7}, 1, 7, 49\}\)[/tex].
- Range of [tex]\( h(x) \)[/tex]: [tex]\(\{-28, -7, 0, -7, -28\} \equiv \{-28, -7, 0\}\)[/tex].
- The range of [tex]\( f(x) \)[/tex] has 5 values.
- The range of [tex]\( g(x) \)[/tex] has 5 values.
- The range of [tex]\( h(x) \)[/tex] has 3 values.
Statement: The range of [tex]\( h(x) \)[/tex] has more values than the other ranges. This is False.
### Step 5: Maximum Values
We compare the maximum values of each function from the given data.
- Maximum of [tex]\( f(x) \)[/tex]: [tex]\(\text{max}(f(x)) = \text{max}(-14, -7, 0, 7, 14) = 14\)[/tex].
- Maximum of [tex]\( g(x) \)[/tex]: [tex]\(\text{max}(g(x)) = \text{max}(\frac{1}{49}, \frac{1}{7}, 1, 7, 49) = 49\)[/tex].
- Maximum of [tex]\( h(x) \)[/tex]: [tex]\(\text{max}(h(x)) = \text{max}(-28, -7, 0, -7, -28) = 0\)[/tex].
Statement: The maximum of [tex]\( g(x) \)[/tex] is greater than the other maximums. This is True.
### Final Answers
Based on the analysis, the true statements are:
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 range of [tex]\( h(x) \)[/tex] has more values than the other ranges. (False)
5. The maximum of [tex]\( g(x) \)[/tex] is greater than the other maximums.
Therefore, select these three options:
- 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.
- The maximum of [tex]\( g(x) \)[/tex] is greater than the other maximums.
