Question
Compare the graphs of f(x)=-√(x) and g(x)=-∛(x). Identify which of the following features is true about the graphs.
a. they have the same domains
b. they have the same ranges
c. they are both increasing
d. they are both decreasing
a. they have the same domains
b. they have the same ranges
c. they are both increasing
d. they are both decreasing
Answers
Answered by
GPT-4o mini
To compare the graphs of \( f(x) = -\sqrt{x} \) and \( g(x) = -\sqrt[3]{x} \), let's analyze their domains and ranges, as well as their behavior (whether they are increasing or decreasing).
### 1. Domains:
- **Domain of \( f(x) = -\sqrt{x} \):**
- The square root function \( \sqrt{x} \) is defined only for \( x \geq 0 \).
- Therefore, the domain of \( f(x) \) is \( [0, \infty) \).
- **Domain of \( g(x) = -\sqrt[3]{x} \):**
- The cube root function \( \sqrt[3]{x} \) is defined for all real numbers.
- Therefore, the domain of \( g(x) \) is \( (-\infty, \infty) \).
**Conclusion for Domains:** \( f(x) \) has a domain of \( [0, \infty) \) and \( g(x) \) has a domain of \( (-\infty, \infty) \). **They do not have the same domains.**
### 2. Ranges:
- **Range of \( f(x) = -\sqrt{x} \):**
- The square root function outputs values starting from 0 upwards, \( \sqrt{x} \geq 0 \).
- Since \( f(x) \) is negative, \( f(x) \) outputs values from \( -\infty \) to 0.
- Therefore, the range of \( f(x) \) is \( (-\infty, 0] \).
- **Range of \( g(x) = -\sqrt[3]{x} \):**
- The cube root function can take any real value, so \( \sqrt[3]{x} \) can be any real number.
- Hence, \( g(x) \) also produces all negative values as \( -\sqrt[3]{x} \) can take any negative value, as well as zero.
- Therefore, the range of \( g(x) \) is \( (-\infty, 0] \).
**Conclusion for Ranges:** Both functions have a range of \( (-\infty, 0] \). **They have the same ranges.**
### 3. Behavior (Increasing/Decreasing):
- **For \( f(x) = -\sqrt{x} \):**
- The function is decreasing since as \( x \) increases, \( -\sqrt{x} \) decreases.
- **For \( g(x) = -\sqrt[3]{x} \):**
- The cube root function increases as \( x \) increases, but since it is negated, \( g(x) \) is also decreasing.
**Conclusion for Behavior:** Both functions are decreasing.
### Summary of Findings:
- a. They have the same domains: **False**
- b. They have the same ranges: **True**
- c. They are both increasing: **False**
- d. They are both decreasing: **True**
Thus, the true statements about the graphs are **b** (they have the same ranges) and **d** (they are both decreasing).
### 1. Domains:
- **Domain of \( f(x) = -\sqrt{x} \):**
- The square root function \( \sqrt{x} \) is defined only for \( x \geq 0 \).
- Therefore, the domain of \( f(x) \) is \( [0, \infty) \).
- **Domain of \( g(x) = -\sqrt[3]{x} \):**
- The cube root function \( \sqrt[3]{x} \) is defined for all real numbers.
- Therefore, the domain of \( g(x) \) is \( (-\infty, \infty) \).
**Conclusion for Domains:** \( f(x) \) has a domain of \( [0, \infty) \) and \( g(x) \) has a domain of \( (-\infty, \infty) \). **They do not have the same domains.**
### 2. Ranges:
- **Range of \( f(x) = -\sqrt{x} \):**
- The square root function outputs values starting from 0 upwards, \( \sqrt{x} \geq 0 \).
- Since \( f(x) \) is negative, \( f(x) \) outputs values from \( -\infty \) to 0.
- Therefore, the range of \( f(x) \) is \( (-\infty, 0] \).
- **Range of \( g(x) = -\sqrt[3]{x} \):**
- The cube root function can take any real value, so \( \sqrt[3]{x} \) can be any real number.
- Hence, \( g(x) \) also produces all negative values as \( -\sqrt[3]{x} \) can take any negative value, as well as zero.
- Therefore, the range of \( g(x) \) is \( (-\infty, 0] \).
**Conclusion for Ranges:** Both functions have a range of \( (-\infty, 0] \). **They have the same ranges.**
### 3. Behavior (Increasing/Decreasing):
- **For \( f(x) = -\sqrt{x} \):**
- The function is decreasing since as \( x \) increases, \( -\sqrt{x} \) decreases.
- **For \( g(x) = -\sqrt[3]{x} \):**
- The cube root function increases as \( x \) increases, but since it is negated, \( g(x) \) is also decreasing.
**Conclusion for Behavior:** Both functions are decreasing.
### Summary of Findings:
- a. They have the same domains: **False**
- b. They have the same ranges: **True**
- c. They are both increasing: **False**
- d. They are both decreasing: **True**
Thus, the true statements about the graphs are **b** (they have the same ranges) and **d** (they are both decreasing).