Answer :
Let's analyze each of the given statements about the function [tex]\( y = \left(\frac{1}{3}\right)^x \)[/tex] one by one:
1. The function is increasing:
To determine whether the function is increasing or decreasing, we need to look at the base of the exponential function. The function [tex]\( y = a^x \)[/tex] is increasing if [tex]\( a > 1 \)[/tex] and decreasing if [tex]\( 0 < a < 1 \)[/tex]. In our case, the base is [tex]\( \frac{1}{3} \)[/tex], which is less than 1 but greater than 0. Hence, the function is decreasing for all real [tex]\( x \)[/tex].
Conclusion: The statement "The function is increasing" is false.
2. The function is decreasing:
As discussed in the previous point, because [tex]\( 0 < \frac{1}{3} < 1 \)[/tex], the function [tex]\( y = \left(\frac{1}{3}\right)^x \)[/tex] is indeed a decreasing function for all real [tex]\( x \)[/tex].
Conclusion: The statement "The function is decreasing" is true.
3. The [tex]\( x \)[/tex]-intercept is [tex]\( (1,0) \)[/tex]:
To find the [tex]\( x \)[/tex]-intercept, we need to set [tex]\( y = 0 \)[/tex] and solve for [tex]\( x \)[/tex]. However, the function [tex]\( y = \left(\frac{1}{3}\right)^x \)[/tex] is never equal to zero for any real [tex]\( x \)[/tex] because the exponential function with a positive base never actually reaches zero; it only gets closer and closer to zero as [tex]\( x \to \infty \)[/tex].
Conclusion: The statement "The [tex]\( x \)[/tex]-intercept is [tex]\( (1,0) \)[/tex]" is false.
4. The [tex]\( y \)[/tex]-intercept is [tex]\( (0,1) \)[/tex]:
To find the [tex]\( y \)[/tex]-intercept, we set [tex]\( x = 0 \)[/tex] and solve for [tex]\( y \)[/tex]. Plugging [tex]\( x = 0 \)[/tex] into the function [tex]\( y = \left(\frac{1}{3}\right)^x \)[/tex] gives us:
[tex]\[ y = \left(\frac{1}{3}\right)^0 = 1. \][/tex]
Therefore, the [tex]\( y \)[/tex]-intercept is indeed [tex]\( (0,1) \)[/tex].
Conclusion: The statement "The [tex]\( y \)[/tex]-intercept is [tex]\( (0,1) \)[/tex]" is true.
5. The range of the function is all real numbers:
The range of the exponential function [tex]\( y = \left(\frac{1}{3}\right)^x \)[/tex] is the set of all possible output values. Since [tex]\( \left(\frac{1}{3}\right)^x \)[/tex] is always a positive number regardless of the value of [tex]\( x \)[/tex], the range is [tex]\( (0, \infty) \)[/tex], not all real numbers.
Conclusion: The statement "The range of the function is all real numbers" is false.
To summarize:
- The function is increasing: False
- The function is decreasing: True
- The [tex]\( x \)[/tex]-intercept is [tex]\( (1,0) \)[/tex]: False
- The [tex]\( y \)[/tex]-intercept is [tex]\( (0,1) \)[/tex]: True
- The range of the function is all real numbers: False
1. The function is increasing:
To determine whether the function is increasing or decreasing, we need to look at the base of the exponential function. The function [tex]\( y = a^x \)[/tex] is increasing if [tex]\( a > 1 \)[/tex] and decreasing if [tex]\( 0 < a < 1 \)[/tex]. In our case, the base is [tex]\( \frac{1}{3} \)[/tex], which is less than 1 but greater than 0. Hence, the function is decreasing for all real [tex]\( x \)[/tex].
Conclusion: The statement "The function is increasing" is false.
2. The function is decreasing:
As discussed in the previous point, because [tex]\( 0 < \frac{1}{3} < 1 \)[/tex], the function [tex]\( y = \left(\frac{1}{3}\right)^x \)[/tex] is indeed a decreasing function for all real [tex]\( x \)[/tex].
Conclusion: The statement "The function is decreasing" is true.
3. The [tex]\( x \)[/tex]-intercept is [tex]\( (1,0) \)[/tex]:
To find the [tex]\( x \)[/tex]-intercept, we need to set [tex]\( y = 0 \)[/tex] and solve for [tex]\( x \)[/tex]. However, the function [tex]\( y = \left(\frac{1}{3}\right)^x \)[/tex] is never equal to zero for any real [tex]\( x \)[/tex] because the exponential function with a positive base never actually reaches zero; it only gets closer and closer to zero as [tex]\( x \to \infty \)[/tex].
Conclusion: The statement "The [tex]\( x \)[/tex]-intercept is [tex]\( (1,0) \)[/tex]" is false.
4. The [tex]\( y \)[/tex]-intercept is [tex]\( (0,1) \)[/tex]:
To find the [tex]\( y \)[/tex]-intercept, we set [tex]\( x = 0 \)[/tex] and solve for [tex]\( y \)[/tex]. Plugging [tex]\( x = 0 \)[/tex] into the function [tex]\( y = \left(\frac{1}{3}\right)^x \)[/tex] gives us:
[tex]\[ y = \left(\frac{1}{3}\right)^0 = 1. \][/tex]
Therefore, the [tex]\( y \)[/tex]-intercept is indeed [tex]\( (0,1) \)[/tex].
Conclusion: The statement "The [tex]\( y \)[/tex]-intercept is [tex]\( (0,1) \)[/tex]" is true.
5. The range of the function is all real numbers:
The range of the exponential function [tex]\( y = \left(\frac{1}{3}\right)^x \)[/tex] is the set of all possible output values. Since [tex]\( \left(\frac{1}{3}\right)^x \)[/tex] is always a positive number regardless of the value of [tex]\( x \)[/tex], the range is [tex]\( (0, \infty) \)[/tex], not all real numbers.
Conclusion: The statement "The range of the function is all real numbers" is false.
To summarize:
- The function is increasing: False
- The function is decreasing: True
- The [tex]\( x \)[/tex]-intercept is [tex]\( (1,0) \)[/tex]: False
- The [tex]\( y \)[/tex]-intercept is [tex]\( (0,1) \)[/tex]: True
- The range of the function is all real numbers: False
