Answered

IDNLearn.com is designed to help you find the answers you need quickly and easily. Ask your questions and get detailed, reliable answers from our community of knowledgeable experts.

Identify the domain and range of the function.

[tex]\[ y = 3 \cdot 5^x \][/tex]

The domain of this function is [tex]\([ -\infty, \infty ]\)[/tex].

The range of this function is [tex]\((0, \infty)\)[/tex].

Sagot :

GinnyAnsweridn
Let's carefully analyze the function [tex]\( y = 3 \cdot 5^x \)[/tex] to determine its domain and range.

### Domain

The domain of a function is the set of all possible input values (x-values) that the function can accept.

For the function [tex]\( y = 3 \cdot 5^x \)[/tex]:
- Exponential functions of the form [tex]\( a \cdot b^x \)[/tex] (where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants and [tex]\( b > 0 \)[/tex]) accept any real number as the input [tex]\( x \)[/tex].
- This is because exponential functions are defined for all real values of [tex]\( x \)[/tex]; there are no restrictions on [tex]\( x \)[/tex] that would make the function undefined.

Thus, the domain of the function [tex]\( y = 3 \cdot 5^x \)[/tex] is all real numbers, which can be written as:
[tex]\[ (-\infty, \infty) \][/tex]

### Range

The range of a function is the set of all possible output values (y-values) that the function can produce.

For the function [tex]\( y = 3 \cdot 5^x \)[/tex]:
- Since [tex]\( 5^x \)[/tex] is an exponential function, [tex]\( 5^x \)[/tex] is always positive for any real number [tex]\( x \)[/tex]. Exponential functions never produce zero or negative values.
- The factor of 3 is a constant multiplier. Therefore, multiplying a positive number [tex]\( 5^x \)[/tex] by 3 still results in a positive number.
- [tex]\( 3 \cdot 5^x \)[/tex] will be positive no matter what the value of [tex]\( x \)[/tex] is.
- As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( 5^x \)[/tex] approaches 0, but never actually reaches 0. Thus, [tex]\( 3 \cdot 5^x \)[/tex] approaches 0 but remains positive.
- As [tex]\( x \)[/tex] approaches positive infinity, [tex]\( 5^x \)[/tex] grows without bound, and so does [tex]\( 3 \cdot 5^x \)[/tex].

Therefore, the function [tex]\( y = 3 \cdot 5^x \)[/tex] produces all positive real numbers, and its range can be written as:
[tex]\[ (0, \infty) \][/tex]

In conclusion:
- The domain of [tex]\( y = 3 \cdot 5^x \)[/tex] is [tex]\( (-\infty, \infty) \)[/tex].
- The range of [tex]\( y = 3 \cdot 5^x \)[/tex] is [tex]\( (0, \infty) \)[/tex].
We appreciate your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. IDNLearn.com is your source for precise answers. Thank you for visiting, and we look forward to helping you again soon.