Welcome to Westonci.ca, where your questions are met with accurate answers from a community of experts and enthusiasts. Join our platform to get reliable answers to your questions from a knowledgeable community of experts. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
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.
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.
