Discover answers to your questions with Westonci.ca, the leading Q&A platform that connects you with knowledgeable experts. Get expert answers to your questions quickly and accurately from our dedicated community of professionals. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our 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.
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.
