Westonci.ca connects you with experts who provide insightful answers to your questions. Join us today and start learning! Our platform connects you with professionals ready to provide precise answers to all your questions in various areas of expertise. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
Sagot :
Let's analyze the exponential function [tex]\( h(x) = 125^x \)[/tex] step-by-step, and address the questions one by one.
### Domain of [tex]\( h(x) = 125^x \)[/tex]
The domain of a function refers to all the possible input values (x-values) that the function can take.
For exponential functions of the form [tex]\( b^x \)[/tex] where [tex]\( b \)[/tex] is a positive real number (in this case, [tex]\( b = 125 \)[/tex]), the exponent [tex]\( x \)[/tex] can be any real number. This is because raising a positive number to any real power (whether positive, negative, or zero) results in a well-defined real number.
Thus, the domain of [tex]\( h(x) = 125^x \)[/tex] is:
[tex]\[ (-\infty, \infty) \][/tex]
### Range of [tex]\( h(x) = 125^x \)[/tex]
The range of a function refers to all the possible output values (y-values) that the function can produce.
For [tex]\( h(x) = 125^x \)[/tex], it is important to understand the behavior of the exponential function:
- When [tex]\( x \)[/tex] is a large positive number, [tex]\( 125^x \)[/tex] becomes very large.
- When [tex]\( x \)[/tex] is zero, [tex]\( 125^0 = 1 \)[/tex].
- When [tex]\( x \)[/tex] is a large negative number, [tex]\( 125^x \)[/tex] gets very close to 0 but never actually reaches 0.
Since [tex]\( 125^x \)[/tex] is always positive for any real [tex]\( x \)[/tex] and can take values from just above 0 to infinitely large, the range of [tex]\( h(x) = 125^x \)[/tex] is:
[tex]\[ (0, \infty) \][/tex]
### Behavior as [tex]\( x \)[/tex] Decreases
As [tex]\( x \)[/tex] decreases, we are considering the behavior of [tex]\( 125^x \)[/tex] for smaller and smaller values of [tex]\( x \)[/tex]:
- For example, if [tex]\( x \)[/tex] decreases from 1 to 0, [tex]\( 125^x \)[/tex] decreases from 125 to 1.
- If [tex]\( x \)[/tex] decreases from 0 to -1, [tex]\( 125^x \)[/tex] decreases from 1 to [tex]\( \frac{1}{125} \)[/tex].
In general, for a base [tex]\( b > 1 \)[/tex], [tex]\( b^x \)[/tex] will decrease as [tex]\( x \)[/tex] decreases. Therefore, as [tex]\( x \)[/tex] decreases, [tex]\( h(x) \)[/tex] also decreases.
### Behavior as [tex]\( x \)[/tex] Increases
On the other hand, as [tex]\( x \)[/tex] increases, we are considering the behavior of [tex]\( 125^x \)[/tex] for larger and larger values of [tex]\( x \)[/tex]:
- For example, if [tex]\( x \)[/tex] increases from -1 to 0, [tex]\( 125^x \)[/tex] increases from [tex]\( \frac{1}{125} \)[/tex] to 1.
- If [tex]\( x \)[/tex] increases from 0 to 1, [tex]\( 125^x \)[/tex] increases from 1 to 125.
In general, for a base [tex]\( b > 1 \)[/tex], [tex]\( b^x \)[/tex] will increase as [tex]\( x \)[/tex] increases. Therefore, as [tex]\( x \)[/tex] increases, [tex]\( h(x) \)[/tex] also increases.
### Summary
To summarize our findings:
- The domain of [tex]\( h(x) = 125^x \)[/tex] is [tex]\( (-\infty, \infty) \)[/tex].
- The range of [tex]\( h(x) = 125^x \)[/tex] is [tex]\( (0, \infty) \)[/tex].
- As [tex]\( x \)[/tex] decreases, [tex]\( h(x) \)[/tex] decreases.
- As [tex]\( x \)[/tex] increases, [tex]\( h(x) \)[/tex] increases.
This gives us a comprehensive understanding of the behavior and characteristics of the exponential function [tex]\( h(x) = 125^x \)[/tex].
### Domain of [tex]\( h(x) = 125^x \)[/tex]
The domain of a function refers to all the possible input values (x-values) that the function can take.
For exponential functions of the form [tex]\( b^x \)[/tex] where [tex]\( b \)[/tex] is a positive real number (in this case, [tex]\( b = 125 \)[/tex]), the exponent [tex]\( x \)[/tex] can be any real number. This is because raising a positive number to any real power (whether positive, negative, or zero) results in a well-defined real number.
Thus, the domain of [tex]\( h(x) = 125^x \)[/tex] is:
[tex]\[ (-\infty, \infty) \][/tex]
### Range of [tex]\( h(x) = 125^x \)[/tex]
The range of a function refers to all the possible output values (y-values) that the function can produce.
For [tex]\( h(x) = 125^x \)[/tex], it is important to understand the behavior of the exponential function:
- When [tex]\( x \)[/tex] is a large positive number, [tex]\( 125^x \)[/tex] becomes very large.
- When [tex]\( x \)[/tex] is zero, [tex]\( 125^0 = 1 \)[/tex].
- When [tex]\( x \)[/tex] is a large negative number, [tex]\( 125^x \)[/tex] gets very close to 0 but never actually reaches 0.
Since [tex]\( 125^x \)[/tex] is always positive for any real [tex]\( x \)[/tex] and can take values from just above 0 to infinitely large, the range of [tex]\( h(x) = 125^x \)[/tex] is:
[tex]\[ (0, \infty) \][/tex]
### Behavior as [tex]\( x \)[/tex] Decreases
As [tex]\( x \)[/tex] decreases, we are considering the behavior of [tex]\( 125^x \)[/tex] for smaller and smaller values of [tex]\( x \)[/tex]:
- For example, if [tex]\( x \)[/tex] decreases from 1 to 0, [tex]\( 125^x \)[/tex] decreases from 125 to 1.
- If [tex]\( x \)[/tex] decreases from 0 to -1, [tex]\( 125^x \)[/tex] decreases from 1 to [tex]\( \frac{1}{125} \)[/tex].
In general, for a base [tex]\( b > 1 \)[/tex], [tex]\( b^x \)[/tex] will decrease as [tex]\( x \)[/tex] decreases. Therefore, as [tex]\( x \)[/tex] decreases, [tex]\( h(x) \)[/tex] also decreases.
### Behavior as [tex]\( x \)[/tex] Increases
On the other hand, as [tex]\( x \)[/tex] increases, we are considering the behavior of [tex]\( 125^x \)[/tex] for larger and larger values of [tex]\( x \)[/tex]:
- For example, if [tex]\( x \)[/tex] increases from -1 to 0, [tex]\( 125^x \)[/tex] increases from [tex]\( \frac{1}{125} \)[/tex] to 1.
- If [tex]\( x \)[/tex] increases from 0 to 1, [tex]\( 125^x \)[/tex] increases from 1 to 125.
In general, for a base [tex]\( b > 1 \)[/tex], [tex]\( b^x \)[/tex] will increase as [tex]\( x \)[/tex] increases. Therefore, as [tex]\( x \)[/tex] increases, [tex]\( h(x) \)[/tex] also increases.
### Summary
To summarize our findings:
- The domain of [tex]\( h(x) = 125^x \)[/tex] is [tex]\( (-\infty, \infty) \)[/tex].
- The range of [tex]\( h(x) = 125^x \)[/tex] is [tex]\( (0, \infty) \)[/tex].
- As [tex]\( x \)[/tex] decreases, [tex]\( h(x) \)[/tex] decreases.
- As [tex]\( x \)[/tex] increases, [tex]\( h(x) \)[/tex] increases.
This gives us a comprehensive understanding of the behavior and characteristics of the exponential function [tex]\( h(x) = 125^x \)[/tex].
We appreciate your visit. Hopefully, the answers you found were beneficial. Don't hesitate to come back for more information. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.
