Answer :
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].
